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I 


ALTERNATING-CURRENT  MACHINES 


BEING   THE    SECOND    VOLUME  OF 

DYNAMO   ELECTRIC   MACHINERY; 

ITS    CONSTRUCTION,    DESIGN, 
AND    OPERATION 


BY 

SAMUEL    SHELDON,    A.M.,  PH.D. 
\( 

PROFESSOR    OF    PHYSICS    AND    ELECTRICAL    ENGINEERING    AT    THE    POLYTECHNIC 
INSTITUTE    OF    BROOKLYN,    MEMBER    OF   THE    AMERICAN    INSTITUTE 
OF   ELECTRICAL    ENGINEERS,    FELLOW    OF    THE    AMERICAN 
ASSOCIATION    FOR    THE    ADVANCEMENT    OF    SCIENCE, 
AND    FELLOW    OF  THE    AMERICAN    ELECTRO- 
THERAPEUTIC    ASSOCIATION 


HOBART    MASON,  B.S.,  E.E. 

ASSISTANT    IN    ELECTRICAL    ENGINEERING    AT    THE    POLYTECHNIC  INSTITUTE 

OF    BROOKLYN,    AND    ASSOCIATE    OF   THE    AMERICAN    INSTITUTE 

OF   ELECTRICAL   ENGINEERS 

SIXTH   EDITION 


NEW   YORK: 

D.   VAN    NOSTRAND    COMPANY 
23  MURRAY  AND  27  WARREN  STS. 

LONDON: 

CROSBY    LOCKWOOD    &   SON 

7  STATIONERS'  HALL  COURT,  LUDGATE  HILL 

10 


OF  THE 

DIVERSITY   I 


VOV 

sj- 
'7°7 


COPYRIGHT,  1902,  BY 
D.  VAN   NOSTRAND  COMPANY 


TYPOGRAPHY  BY  C.  J.  PETERS  &  SON. 


PRESSWORK  BY  THE  F.  H.  GILSON  COMPANY 
BOSTON,  MASS.,  U.  S.  A. 


PREFACE. 


THIS  book,  like  its  companion  volume  on  Direct  Current 
Machines,  is  primarily  intended  as  a  text-book  for  use  in 
technical  educational  institutions.  It  is  hoped  and  be- 
lieved that  it  will  also  be  of  use  to  those  electrical,  civil, 
mechanical,  and  hydraulic  engineers  who  are  not  perfectly 
familiar  with  the  subject  of  Alternating  Currents,  but  whose 
work  leads  them  into  this  field.  It  is  furthermore  intended 
for  use  by  those  who  are  earnestly  studying  the  subject 
by  themselves,  and  who  have  previously  acquired  some 
proficiency  in  mathematics. 

There  are  several  methods  of  treatment  of  alternating- 
current  problems.  Any  point  is  susceptible  of  demonstra- 
tion by  each  of  the  methods.  The  use  of  all  methods  in 
connection  with  every  point  leads  to  complexity,  and  is 
undesirable  in  a  book  of  this  character.  In  each  case  that 
method  has  been  chosen  which  was  deemed  clearest  and 
most  concise.  No  use  has  been  made  of  the  method  of 
complex  imaginary  numbers. 

A  thorough  understanding  of  what  takes  place  in  an 
alternating-current  circuit  is  not  to  be  easily  acquired.  It 
is  believed,  however,  that  one  who  has  mastered  the  first 
four  chapters  of  this  book  will  be  able  to  solve  any  practi- 
cal problem  concerning  the  relations  which  exist  between 
power,  electro-motive  forces,  currents,  and  their  phases  in 

172114 


iv  PREFACE. 

series  or  multiple  alternating-current  circuits  containing 
resistance,  capacity,  and  inductance. 

The  next  four  chapters  are  devoted  to  the  construction, 
principle  of  operation,  and  behavior  of  the  various  types  of 
alternating-current  machines.  Only  American  machines 
have  been  considered. 

A  large  amount  of  alternating-current  apparatus  is  used 
in  connection  with  plants  for  the  long-distance  transmission 
of  power.  This  subject  is  treated  in  the  ninth  chapter. 
The  last  chapter  gives  directions  for  making  a  variety  of 
tests  on  alternating-current  circuits  and  apparatus. 

No  apology  is  necessary  for  the  introduction  of  cuts  and 
material  supplied  by  the  various  manufacturing  companies. 
The  information  and  ability  of  their  engineers,  and  the  taste 
and  skill  of  their  artists,  are  unsurpassed,  and  the  informa- 
tion supplied  by  them  is  not  available  from  other  sources. 
For  their  courteous  favors  thanks  is  hereby  given. 


CONTENTS. 


CHAPTER  PAGE 

I.  PROPERTIES  OF  ALTERNATING  CURRENTS i 

II.  SELF-INDUCTION       -.    .   • 15 

III.  CAPACITY     .     .    *    .    ...    ...    .    .    .    .    .         .    .    .  29 

IV.  PROBLEMS  ON  ALTERNATING-CURRENT  CIRCUITS.     ...  44 
V.  ALTERNATORS  .     .     .    .    .    .    .    .    ...     •    .     •    •     •     •  57 

VI.  THE  TRANSFORMER      ...............  92 

VII.  MOTORS    ..;....".    .    .    .    .    .    .    ..    .    ...  141 

VIII.  CONVERTERS 169 

IX.  POWER  TRANSMISSION       182 

X.  TESTS  .  198 


ALTERNATING-CURRENT  MACHINES. 


CHAPTER    I. 

PROPERTIES  OF  ALTERNATING  CURRENTS. 

i.  Definition  of  an  Alternating  Current.  —  An  alter- 
nating current  of  electricity  is  a  current  which  changes 
its  direction  of  flow  at  regularly  recurring  intervals. 
Between  these  intervals  the  value  of  the  current  may 
vary  in  any  way.  In  usual  practice,  the  value  varies  with 
some  regularity  from  zero  to  a  maximum,  and  decreases 
with  the  same  regularity  to  zero,  then  to  an  equal  max- 
imum in  the  other  direction,  and  finally  to  zero  again.  In 
practice,  too,  the  intervals  of  current  flow  are  very  short, 
ranging  from  ^  to  ^|¥  second. 


2.  Frequency.  —  When,  as  stated  above,  a  current  has 
passed  from  zero  to  a  maximum  in  one  direction,  to  zero, 
to  a  maximum  in  the  other  direction,  and  finally  to  zero 
again,  it  is  said  to  have  completed  one  cycle.  That  is  to 
say,  it  has  returned  to  the  condition  in  which  it  was  first 
considered,  both  as  to  value  and  as  to  direction,  and  is 
prepared  to  repeat  the  process  described,  making  a  second 
cycle.  It  should  be  noted  that  it  takes  two  alternations 
to  make  one  cycle.  The  tilde  (  ~~  )  is  frequently  used  to 
denote  cycles. 


2  ALTERNATING-CURRENT    MACHINES. 

The  term  frequency  is  applied  to  the  number  of  cycles 
completed  in  a  unit  time,  i.e.,  in  one  second.  Occasionally 
the  word  alternations  is  used,  in  which  case,  unless  other- 
wise specified,  the  number  of  alternations  per  minute  is 
meant.  Thus  the  same  current  is  spoken  of  as  having  a 
frequency  of  25,  or  as  having  3000  alternations.  The  use 
of  the  word  alternations  is  condemned  by  good  practice. 
In  algebraic  notation  the  letter  f  usually  stands  for  the 
frequency. 

The  frequency  of  a  commercial  alternating  current 
depends  upon  the  work  expected  of  it.  For  power  a 
low  frequency  is  desirable,  particularly  for  converters. 
The  great  Niagara  power  plant  uses  a  frequency  of  25. 
Lamps,  however,  are  operated  satisfactorily  only  on  fre- 
quencies of  50  or  more.  Early  machines  had  higher 
frequencies, —  125  and  133  (16,000  alternations)  being 
usual,  — but  these  are  almost  entirely  abandoned  because 
of  their  increased  losses  and  their  unadaptability  to  the 
operation  of  motors  and  similar  apparatus. 

In  the  Report  of  the  Committee  on  Standardization  of 
the  American  Institute  of  Electrical  Engineers  is  the 
following :  "  In  alternating-current  circuits,  the  follow- 
ing approximate  frequencies  are  recommended  as  de- 
sirable : 

25  or  30  40  60  120 

"These  frequencies  are  already  in  extensive  use,  and 
it  is  deemed  advisable  to  adhere  to  them  as  closely  as 
possible." 

The  frequency  of  an  alternating  current  is  always  that 
of  the  E.M.F.  producing  it.  To  find  the  frequency  of  the 
pressure  or  the  current  produced  by  any  alternating-cur- 


PROPERTIES   OF   ALTERNATING   CURRENTS. 


rent   generator,  if    V  be  the   number  of   revolutions   per 
minute,  and  /  be  the  number  of  pairs  of  poles,  then 

/->£; 

3.   Wave-shape If,    in   an    alternating    current,    the 

instantaneous  values  of  current  be  taken  as  ordinates,  and 
time    be    the    abscissae,    a 
curve,  as  in  Fig.  i,  may  be 
developed.     The  length  of 
the  abscissa  for  one  com- 


Fig.   i. 


plete    cycle    is—  seconds. 

Imagine  a  small  cylinder, 
Fig.  2,  carried  on  one  end  of  a  wire,  and  rotated  uniformly 
about  the  other  end  in  a  vertical  plane.  Imagine  a  hori- 
zontal beam  of  parallel  rays  of  light  to  be  parallel  to  the 
plane  of  rotation,  and  to  cast  a  shadow  of  the  cylinder  on 


Fig. 


a  plane  screen  perpendicular  to  the  rays.  The  shadow 
will  move  up  and  down,  passing  from  the  top  of  its  travel 
to  the  bottom  in  a  half  revolution,  and  from  the  bottom 


4  ALTERNATING-CURRENT   MACHINES. 

back  to  the  top  in  another  half  revolution  with  a  perfect 
harmonic  motion.  Now  imagine  the  screen  to  be  moved 
horizontally  in  its  own  plane  with  a  uniform  motion,  and 
the  positions  of  the  shadow  suitably  recorded  on  it, — as 

on  sensitized  paper  or  on 
a  photographic  film,  a 
slotted  screen  protecting 
all  but  the  desired  portion 
from  exposure.  Then  the 
trace  of  the  shadow  will 
be  as  in  Fig.  3.  The 
abscissae  of  this  curve 

may  be  taken  as  time,  as  in  the  preceding  curve,  the  ab- 
scissa of  one  complete  cycle  being  the  time  in  seconds  of 
one  revolution.  Or,  with  equal  relevancy,  the  abscissae 
may  be  expressed  in  degrees.  Consider  the  cylinder  to  be 
in  a  zero  position  when  the  radius  to  which  it  is  attached 
is  horizontal.  Then  the  abscissa  of  any  point  is  the  angle 
which  must  be  turned  through  in  order  that  the  cylinder 
may  cast  its  shadow  at  that  point.  In  this  case  the  abscissa 
of  a  complete  cycle  will  be  360°,  or  2?r.  Consideration  of 
the  manner  in  which  the  curve  has  been  formed  shows 
that  the  ordinate  of  any  point  is  proportional  to  the  sine 
of  the  abscissa  of  that  point,  expressed  in  degrees.  Hence 
this  is  called  a  sinusoid  or  sine  curve. 

If  the  maximum  ordinate  of  this  curve  be  taken  as  Em, 
and  time  be  considered  to  commence  at  the  beginning  of 
any  cycle,  then  the  ordinate  E'  at  any  time  t  seconds  later 

will  be 

Er  =  Em  sin  2  TT//, 

which  is  equivalent  to  neglecting  all  those  intervals  of 
time  corresponding  to  whole  cycles,  and  considering  only 


PROPERTIES   OF   ALTERNATING   CURRENTS.        5 

the   time   elapsed    since    the    end  of   the  last    completed 
cycle. 

As  a  numerical  example :  In  an  alternating-current 
circuit  of  45^  and  a  maximum  voltage  of  100,  what 
will  be  the  pressure  at  2!  seconds  after  the  beginning  of 
a  cycle  ? 

E'  =  100  sin  (2  TT  X  45  X  2.125) 

E'  i 
=  sin  iqi.2  c  TT  =  sin  1.2  c  TT  =  —  — —  > 

100  ^2 

whence 

E'  =  —  70.7  volts. 

Since  the  ordinates  of  the  curve  may  represent  either 
current  or  pressure,  the  expression 

/'  =  jm  sin  27T/? 
is  equally  true. 

The  ideal  pressure  curve  from  an  alternator  is  sin- 
usoidal. Commercial  alternators,  however,  do  not  gene- 
rate true  sinusoidal  pressures.  But  the  sine  curve  can 
be  treated  with  relative  simplicity,  and  the  curves  of 
practice  approximate  so  closely  to  the  sine  form,  that 
mathematical  deductions  based  on  sine  curves  can  with 
propriety  be  applied  to  those  of  practice.  Two  of  these 
actual  curves  are  shown  in  Fig.  4. 

The  shape  of  the  pressure  curve. is  affected  by  irregular 
distribution  of  the  magnetic  flux.  Also  uneven  angular 
velocity  of  the  generator  will  distort  the  wave-shape, 
making  it,  relative  to  the  true  curve,  lower  in  the  slow 
spots  and  higher  in  the  fast  ones.  Again,  the  magnetic 
reluctance  of  the  armature  may  vary  in  different  angular 
positions,  particularly  if  the  inductors  are  laid  in  a  few 
large  slots.  This  would  cause  a  periodic  variation  in  the 


ALTERNATING-CURRENT    MACHINES. 


reluctance  of  the  whole  magnetic  circuit  and  a  correspond- 
ing pulsation  of  the  total  magnetic  flux.  All  these  influ- 
ences operate  at  open  circuit  as  well  as  under  load. 


E  M.F.  CURVE 
3  PHASE 
40  POLE 
20OO  K.W. 

26 
FULLY  LOADED 


E.M.F.  CURVE 
SINGLE  PHASE 

8  POLE 
500  WATTS 

125 
NOT  LOADED 


Fig.  4. 


There  are  two  other  causes  which  act  to  distort  the 
wave-shape  only  when  under  load.  For  any  separately 
excited  generator,  a  change  in  the  resistance  or  apparent 
resistance  of  the  external  circuit  will  cause  a  change  in  the 


PROPERTIES  OF  ALTERNATING  CURRENTS.   / 

terminal  voltage  of  the  machine.  As  is  explained  later, 
the  apparent  resistance  (impedance)  of  a  circuit  to  alter- 
nating currents  depends  upon  the  permeability  of  the  iron 
adjacent  to  the  circuit.  Permeability  changes  with  mag- 
netization. Now,  because  an  alternating  current  is  flow- 
ing, the  magnetization  changes  with  the  changing  values 
of  current.  This,  by  varying  the  permeability,  sets  up  a 
pulsation  in  the  impedance  and  affects  the  terminal  volt- 
age of  the  machine,  periodically  distorting  the  wave  of 
pressure  from  the  true  sine. 

There  are  cases  of  synchronously  pulsating  resistances. 
The  most  common  is  that  of  the  alternating  arc.  With 
the  same  arc  the  apparent  resistance  of  the  arc  varies  in- 
versely as  the  current.  So  when  operated  by  alternating 
currents,  the  resistance  of  a  circuit  of  arc  lamps  varies  syn- 
chronously, and  distorts  the  pressure  waveshape  in  a 
manner  analogous  to  the  above. 

Summing  up,  the  wave-shape  of  pressure  may  be  dis- 
torted :  At  open  circuit  as  well  as  tinder  load ;  by  lack  of 
uniformity  of  magnetic  distribution,  by  pulsating  of  mag- 
netic field,  by  variation  in  angular  velocity  of  armature  ; 
and  under  load  only  ;  by  pulsation  of  impedance,  by  pulsa- 
tion of  resistance.  And  the  effects  of  any  or  all  may  be 
superimposed. 

4.   Effective  Values  of  E.M.F.  and  of  Current One 

ampere  of  alternating  current  is  a  current  of  such  instan- 
taneous values  as  to  have  the  same  heating  effect  in  a  con- 
ductor as  one  ampere  of  direct  current.  This  somewhat 
arbitrary  definition  probably  arose  from  the  fact  that  alter- 
nating currents  were  first  commercially  employed  in  light- 
ing circuits,  where  their  utility  was  measured  by  the  heat 


8 


ALTERNATING-CURRENT    MACHINES. 


they  produced  in  the  filaments  ;  and  further  from  the  fact 
that  the  only  means  then  at  hand  of  measuring  alternating 
currents  were  the  hot-wire  instruments  and  the  electro- 
dynamometer,  either  of  which  gives  the  same  indication 
for  an  ampere  of  direct  current  or  for  what  is  now  called 
an  ampere  of  alternating  current. 

The  heat  produced  in  a  conductor  carrying  a  current  is 
proportional  to  the  square  of  the  current.  In  an  alternat- 
ing current,  whose  instantaneous  current  values  vary,  the 
instantaneous  rate  of  heating  is  not  proportional  to  the 
instantaneous  value,  nor  yet  to  the  square  of  the  average 

of  the  current  values,  but  to  the 
square  of  the  instantaneous  cur- 
rent value.  And  so  the  average 
heating  effect  is  proportional  to 

\'    the  mean  of  the  squares  of  the 
instantaneous  currents. 

The  average  current  of  a  sinu- 
soidal wave  of  alternating  current,  whose  maximum  value 
is  7W,  is  equal  to  the  area  of  one  lobe  of  the  curve,  Fig.  5, 
divided  by  its  base  line  ?r.  Thus 


Fig.  5. 


7     —  ' 


jr 


7,,  sin  OttO 


But  the  heating  value  of  such  a  current  varies,  as 

f  7W2  si 
Jo 


sn 


72  = 


-      sn 
4 


The  square  root  of  this  quantity  is  called  the  effective 
value  of  the  current,  7  =  — =•     This  has  the  same  heating 

V2 


PROPERTIES    OF  ALTERNATING   CURRENTS.        9 

effect  as  a  direct  current  /,  and  the  effective  values  are 
always  referred  to  unless  expressly  stated  otherwise. 
Alternating-current  ammeters  are  designed  to  read  in 
effective  amperes. 

Since  current  is  dependent  upon  the  pressure,  the 
resistance  or  apparent  resistance  of  a  circuit  remain- 
ing constant,  it  is  obvious  that  if  /  =  -^  then  does 

E  2       V2 

also  E  —     "1-    Likewise  if  average  7  =  -  Im  then  does  also 

V2  «• 

average  E  =  -  Em.     Or  these  may  be  demonstrated  in  a 

7T 

manner  analogous  to  the  above. 

The  maximum  value  of  pressure  is  frequently  referred 
to  in  designing  alternator  armatures,  and  in  calculating 
dielectric  strength  of  insulation.  There  have  arisen  vari- 
ous ways  of  indicating  that  effective  values  are  meant, 
for  instance,  the  expressions,  sq.  root  of  mean  sq.,  V^2, 
Vmean  square.  In  England  the  initials  R.M.S.  are  fre- 
quently used  for  root  mean  square. 

~,  Effective  E.M.F. 

Ihe  ratio  — —  ^  n/r  ^    is    called    the  form-factor, 

Average  E.M.F. 

since    its   value    depends    upon 
the  shape  of  the  pressure  wave. 
For  the  curve  Fig.  6,  the  form- 
factor  is  unity.     As  a  curve  be- 
comes  more   peaked,  its    form-  Kg.  e. 
factor  increases,  due  to  the  superior  weight  of  the  squares 
of  the  longer  ordinates. 

In  the  sinusoid  the  values  found  above  give 


Form-factor  = 


=  i.n. 


10 


ALTERNATING-CURRENT   MACHINES. 


Probably  no  alternators  give  sine  waves,  but  they  ap- 
proach it  so  nearly  that  the  value  i.n  can  be  used  in  cal- 
culation without  sensible  error. 


IN  PHASE 


5.    Phase The  curves  of  the  pressure  and  the  current 

in  a  circuit  can  be  plotted  together,  with  their  respective 
ordinates  and  common  abscissae,  as  in  Fig.  7.  In  some 

cases  the  zero  and  the 
maximum  values  of  the 
current  curve  will  occur 
at  the  same  abscissae  as 
do  those  values  of  the 
pressure  curve,  as  in  Fig. 
Fig-  7*  7.  In  such  a  case  the 

current  is  said  to  be  in  phase  with  the  pressure.  In  other 
cases  the  current  will  reach  a  maximum  or  a  zero  value  at 
a  time  later  than  the  corresponding  values  of  the  pressure, 
and  since  the  abscissae  are  indifferently  time  or  degrees, 
the  condition  is  represented  in  Fig.  8.  In  such  a  case, 
the  current  is  said  to  be  out  of  phase  with,  and  to  lag  be- 
hind the  pressure.  In 

Still     Other     cases     the  /          .--N^x      LAGGING  CURRENT 

curves  are  placed  as  in 
Fig.  9,  and  the  current 
and  pressure  are  again 
out  of  phase,  but  the 
current  is  said  to  lead  Flg-  8> 

the  pressure.  The  distance  between  the  zero  ordinate  of 
one  sine  curve  and  the  corresponding  zero  ordinate  of 
another,  may  be  measured  in  degrees,  and  is  called  the 
angular  displacement  or  phase  difference.  This  angle  of 
lag  or  of  lead  is  usually  represented  by  <£.  When  one 


PROPERTIES    OF  ALTERNATING   CURRENTS.      II 


LEADING  CURRENT 


RIGHT  ANGLES 


curve  has  its  zero  ordinate  coincident  with  the  maximum 
ordinate  of  the  other,  as  in  Fig.  10,  there  is  a  displacement 
of  a  quarter  cycle  (<£  =  90°),  and  the  curves  are  said  to  be 
at  right  angles.  This 
term  owes  its  origin  to 
the  fact  that  the  radii 
whose  projections  will 
trace  these  curves,  as 
in  §  3,  are  at  right 
angles  to  each  other.  Fig-  9- 

If  the  zero  ordinates  of  the  two  curves  coincide,  but  the 
positive  maximum  of  one  coincides  with  the  negative  maxi- 
mum of  the  other,  as  in 
Fig.  n,  then  <£  =  i8o°> 
and  the  curves  are  in  op- 
posite phase. 

An  alternator  arranged 
to  give  a  single  pressure 
wave  to  a  two- wire  circuit  is 
said  to  be  a  single  phaser, 

and  the  current  in  the  circuit  a  single-phase  current^ 
Some  machines  are  arranged  to  give  pressure  to  two  dis- 
tinct circuits  —  each  of 
which,  considered  alone, 
is  a  single-phase  circuit 
—  but  the  time  of  maxi- 
mum pressure  in  one  is 
the  time  of  zero  pres- 
sure in  the  other,  so 
that  simultaneous  pres- 
sure curves  from  the  two  circuits  take  the  form  of 
Fig.  12.  Such  is  said  to  be  a  two-phase  or  quarter-phase 


2*1° 


OPPOSITE  PHASE 


\ 


Fig.  xi. 


12 


ALTERNATING-CURRENT    MACHINES. 


system,  and  the  generator  is  a  two-phaser.  A  three-phase 
system  theoretically  has  three  circuits  of  two  wires  each. 
The  maximum  positive  pressure  on  any  circuit  is  displaced 
from  that  of  either  of  the  other  circuits  by  1 20°.  As  the 

algebraic  sum  of  the  cur- 
rents in  all  these  circuits 
(if  balanced)  is  at  every  in- 
stant equal  to  zero,  the 
three  return  wireS)  one  on 


each    circuit,    may    be    dis- 
pensed   with,    leaving    but 
TWO  PHASE  three  wjres     The  three  sim- 

Fig.  12. 

ultaneous  curves  of  E.M.F. 

are  shown  in  Fig.  13.  The  term  polyphase  applies  to  any 
system  of  two  or  more  phases.  An  //-phase  system  has  n 
circuits  and  n  pressures  with  successive  phase  differences 

of  - —  degrees. 
n 

6.  Power  in  Alternating-Current  Circuits.  —  With  a  direct- 
current  circuit,  the  power  in  the  circuit  is  equal  to  the 
product  of  the  pressure  in  volts  by  the  current  strength  in 
amperes.  In  an  alternating- 
current  circuit,  the  instan- 
taneous power  is  the  product 
of  the  instantaneous  values 


of     current     strength     and 

pressure.      If    the     current 

and    pressure    are    out    of 

phase    there    will   be    some 

instants  when  the  pressure  will  have  a  positive  value  and 

the  current  a  negative  value  or  vice  versa.     At  such  times 

the  instantaneous  power  will  be  a  negative  quantity,  i.e., 


PROPERTIES   OF   ALTERNATING   CURRENTS.      13 


power  is  being  returned  to  the  generator  by  the  disappear- 

ing magnetic  field  which  had  been  previously  produced  by 

the  current.     This  condition  is  shown  in  Fig.  14,   where 

the  power  curve  has  for  its  ordinates  the  product  of  the 

corresponding  ordinates  of   pressure  and  current.     These 

are  reduced  by  multiplying  by  a  constant  so  as  to  make 

them  of  convenient   size. 

The     circuit,      therefore, 

receives  power  from   the 

generator  and  gives  power 

back  again  in  alternating 

pulsations    having    twice 

the  frequency  of  the  gen- 

erator.    It   is    clear  that 

the    relative     magnitudes  Flg>  I4> 

of  the  negative  and  positive  lobes  of  the  power  curve  will 

vary  for  different    values  of   </>,  even  though  the  original 

curves  maintain  the  same  size  and  shape.      So  it  follows 

that  the  power   in   an   alternating-current    circuit    is    not 

merely  a  function  of  E  and  /,  as  in  direct-current  circuits, 

but  is  a  function  of  E,  /,  and  </>,  and  the  relation  is  deduced 

as  follows  :  — 

Let  the  accent  (')  denote  instantaneous  values.     If  the 
current  lag  by  the  angle  <f>,  then  from  §  3, 

E'  =  Em  sin  a, 
where,  for  convenience, 

a  =  2  IT  ft, 

and  I'  =  Im  sin  (a  —  <£). 

Remembering  that 

j?  r 

E  =  —  -^,    and     /  =  —  ^  (§  4)  the  instantaneous  power, 

V2  V2 


Pf  = 


'  =  2EI  sin  a  sin  (a  - 


14  ALTERNATING-CURRENT    MACHINES. 

But  sin  (a  —  <£)  =  sin  a  cos  <£  —  cos  a  sin  <£, 

so  P'  =  2  El  (sin2  a  cos  <f>  —  sin  a  cos  a  sin  </>). 

Remembering  that    <£  is   a  constant,   the   average   power 
over  1  80°, 

2  El  cos  <f>  r*  .  2^  /sin  <£  /'»•  . 

P  =  —  -    I     sin'2  cu/a  --  -    I     Sill  a  COS  atf  a 

7T  Jo  TT  t/0 

2^/cos<f>ri         i    ,         >      2  ^7  sin  A  fi    ,       > 
=  -  -a  --  Sin  2  a      --  -  Sin2  a     . 

7T  |_2  4  J0  7T  L2  JO 


Should  the  current  lead  the  pressure  by  <£°,  then  the 
leading  equation  would  be 

jPf=  2  El  sin  a  sin  (a  +  <£), 
which  gives  the  same  expression, 

P  =  El  cos  <£, 

which  is  the  general  expression  for  power  in  an  alternating- 
current  circuit. 

Since,  to  get  the  true  power  in  the  circuit,  the  apparent 
power,  or  volt  -amperes,  must  be  multiplied  by  cos  <£,  this 
quantity  is  called  \h.z  power  factor  of  the  circuit.  If  the 
pressure  and  current  are  in  phase,  <£  =  o°,  and  the  power 
factor  is  unity. 


SELF-INDUCTION.  1 5 


CHAPTER   II. 

SELF-INDUCTION. 

7.  Self -Inductance.  —  The  subject  of  inductance  was 
briefly  treated  of  in  §  15,  vol.  i.,  of  this  work  ;  but,  since  it 
is  an  essential  part  of  alternating-current  phenomena,  it 
will  be  discussed  more  fully  in  this  chapter.  When  lines 
of  force  are  cut  by  a  conductor  an  E.M.F.  is  generated  in 
that  conductor  (§  13,  vol.  i.).  A  conductor  carrying  cur- 
rent is  encircled  by  lines  of  force.  When  the  current  is 
first  started  in  such  a  conductor,  these  lines  of  force  must 
be  established.  In  establishing  itself,  each  line  is  con- 
sidered as  having  cut  the  conductor,  or,  what  is  equivalent, 
been  cut  by  the  conductor.  This  notion  of  lines  of  force 
is  a  convenient  fiction,  designed  to  render  an  understand- 
ing of  the  subject  more  easy.  To  account  for  the  E.M.F. 
of  self-induction,  the  encircling  lines  must  be  considered 
as  cutting  the  conductor  which  carries  the  current  that 
establishes  them,  during  their  establishment.  It  may  be 
considered  that  they  start  from  the  axis  of  the  conductor 
at  the  moment  of  starting  the  current  in  the  circuit ;  that 
they  grow  in  diameter  while  the  current  is  increasing ;  that 
they  shrink  in  diameter  when  the  current  is  decreasing; 
and  that  all  their  diameters  reduce  to  zero  upon  stopping 
the  current.  At  any  given  current  strength  the  conductor 
is  surrounded  by  many  circular  lines,  the  circles  having 
various  diameters.  Upon  decreasing  the  strength  those  of 


16  ALTERNATING-CURRENT    MACHINES. 

smaller  diameter  cut  the  conductor  and  disappear  into  a 
point  on  the  axis  of  the  conductor  previous  to  the  cutting 
by  those  of  larger  diameter.  The  number  of  lines  accom- 
panying a  large  current  is  greater  than  the  number  accom- 
panying a  smaller  current. 

The  E.M.F.  of  self-induction  is  always  a  counter  E.M.F. 
By  this  is  meant  that  its  direction  is  such  as  to  tend  to 
prevent  the  change  of  current  which  causes  it.  When  the 
current  is  started  the  self -induced  pressure  tends  to  oppose 
the  flow  of  the  current  and  prevents  its  reaching  its  full 
value  immediately.  When  the  circuit  is  interrupted  the 
E.M.F.  of  self-induction  tends  to  keep  the  current  flowing 
in  the  same  direction  that  it  had  originally. 

8.  Unit  of  Self -Inductance The  self -inductance,  or 

the  coefficient  of  self -induction  of  a  circuit  is  generally  rep- 
resented by  L  or  /,  and  is  that  constant  by  which  the  time 
rate  of  change  of  the  current  in  a  circuit  must  be  multi- 
plied in  order  to  give  the  E.M.F.  induced  in  that  circuit. 
Its  absolute  value  is  numerically  equal  to  the  number  of 
lines  of  force  linked  with  the  circuit,  per  absolute  unit  of 
current  in  the  circuit,  as  is  shown  below.  By  linkages,  or 
number  of  lines  linked  with  a  circuit,  is  meant  the  sum 
of  the  number  of  lines  surrounding  each  portion  of  the 
circuit.  For  instance,  a  coil  of  wire  consisting  of  ten 
turns,  and  threaded  completely  through  by  twelve  lines 
of  force,  is  said  to  have  1 20  linkages. 

The  absolute  unit  of  self-inductance  is  too  small  for 
ordinary  purposes,  and  a  practical  unit,  the  henry,  is  used. 
This  is  io9  times  as  large  as  the  c.  G.  s.  or  absolute  unit. 

The  Paris  electrical  congress  of  1900  adopted  as  the 
unit  of  magnetic  flux  the  maxwell,  and  of  flux  density  the 


SELF-INDUCTION.  17 

gauss.  A  maxwell  is  one  line  of  force.  A  gauss  is  one 
line  of  force  per  square  centimeter.  If  a  core  of  an  electro- 
magnet has  a  transverse  cross-section  of  30  sq.  cm.,  and  is 
uniformly  permeated  with  60,000  lines  of  force,  such  a 
core  may  be  said  to  have  a  flux  of  60,000  maxwells  and  a 
flux  density  of  2000  gausses. 

In  §  13,  vol.  i.,  it  has  been  shown  that  the  pressure  gene- 
rated in  a  coil  of  wire  when  it  is  cut  by  lines  of  force  is 


where  n  is  the  number  of  turns  in  a  coil,  and  where  e  is 
measured  in  c.  G.  s.  units,  <£  in  maxwells,  and  /  in  seconds. 
In  a  simple  case  of  self-induction  the  maxwells  set  up  are 
due  solely  to  the  current  in  the  conductor.  Now  let  K  be 
a  constant,  dependent  upon  the  permeability  of  the  mag- 
netic circuit,  such  that  it  represents  the  number  of  max- 
wells set  up  per  unit  current  in  the  electric  circuit  ;  then, 
indicating  instantaneous  values  by  prime  accents, 

&  =  Kir, 

and  d&  =  Kdi. 

The  E.M.F.  of  self-induction  may  then  be  written 


By  the  definition  of    the  coefficient    of    self-induction, 
whose  c.  G.  s.  value  is  represented  by  /, 


From  the  last  two  equations,  it  is  seen  that  /  =  Kn.  Kn  is 
evidently  the  number  of  linkages  per  absolute  unit  current. 
The  negative  sign  indicates  that  the  pressure  is  counter 
E.M.F, 


18  ALTERNATING-CURRENT  MACHINES. 

In  practical  units, 

F         rdT 
E'=-L~dt- 

A  circuit  having  an  inductance  of  one  henry  will  have  a 
pressure  of  one  volt  induced  in  it  by  a  uniform  change  of 
current  of  one  ampere  per  second. 

9.  Practical  Values  of  Inductances.  —  To  give  the  stu- 
dent an  idea  of  the  values  of  self -inductance  met  with  in 
practice,  a  number  of  examples  are  here  cited. 

A  pair  of  copper  line  wires,  say  a  telephone  pole  line, 
will  have  from  two  to  four  milhenrys  (.002  to  .004  henrys) 
per  mile,  according  to  the  distance  between  them,  the 
larger  value  being  for  the  greater  distance. 

The  secondary  of  an  induction  coil  giving  a  2"  spark  has 
a  resistance  of  about  6000  ohms  and  50  henrys. 

The  secondary  of  a  much  larger  coil  has  30,000  ohms 
and  about  2000  henrys. 

A  telephone  call  bell  with  about  75  ohms  has  1.5  henrys. 

A  coil  found  very  useful  in  illustrative  and  quantitive 
experiments  in  the  alternating-current  laboratory  is  of  the 
following  dimensions.  It  is  wound  on  a  pasteboard  cylinder 
with  wooden  ends,  making  a  spool  8.5  inches  long  and  2 
inches  internal  diameter.  This  is  wound  to  a  depth  of  1.5 
inch  with  No.  16  B.  and  S.  double  cotton-covered  copper 
wire,  there  being  about  3000  turns  in  all.  A  bundle  of 
iron  wires,  16  inches  long,  fits  loosely  in  the  hole  of  the 
spool.  The  resistance  of  the  coil  is  10  ohms,  and  its  in- 
ductance without  the  core  is  0.2  henry.  With  the  iron 
core  in  place  and  a  current  of  about  0.2  ampere,  the  induc- 
tance is  about  1.75  henrys.  This  coil  is  referred  to  again 
in  §  ii. 


SELF-INDUCTION.  19 

The  inductance  of  a  spool  on  the  field  frame  of  a  gene- 
rator is  numerically 


where  <$  is  the  total  flux  from  one  pole,  n  the  number 
of  turns  per  spool,  and  If  the  field  current  of  the  machine. 
It  is  evident  that  the  value  of  L  may  vary  through  a  wide 
range  with  different  machines. 

10.   Things  Which  Influence  the  Magnitude  of  L.  —  If  all 

the  conditions  remain  constant,  save  those  under  considera- 
tion, then  the  self-inductance  of  a  coil  will  vary  :  directly  as 
the  square  of  the  number  of  turns  ;  directly  as  the  linear 
dimension  if  the  coil  changes  its  size  without  changing  its 
shape  ;  and  inversely  as  the  reluctance  of  the  magnetic 
circuit. 

Any  of  the  above  relations  is  apparent  from  the  follow- 
ing equations.  The  numerical  value  of  the  self-induc- 
tance is 

/=  n  —  • 
t 

As  shown  in  Chapter  2,  vol.  i., 

M.M.F.         4  Trni 
<E>  =  —  --  =  -  , 
reluctance          c 

fA 

where  c  is  the  mean  length  in  centimeters  of  the  magnetic 
circuit,  A  its  mean  cross-sectional  area  in  square  centi- 
meters, and  /u,  is  permeability. 

Then,  if  <H  stand  for  the  reluctance, 
n    4  trm  ,    A 


which  is  independent  of  i, 


20  ALTERNATING-CURRENT    MACHINES. 

If,  as  is  generally  the  case,  there  is  iron  in  the  magnetic 
circuit,  it  is  practically  impossible  to  keep  /x  constant  if  any 
of  the  conditions  are  altered  ;  and  it  is  to  be  particularly 
noted,  that  with  iron  in  the  magnetic  circuit,  L  is  by  no 
means  independent  of  /. 

ii.   Growth  of  Current  in  an  Inductive   Circuit.  —  If  a 

constant  E.M.F.  be  applied  to  the  terminals  of  a  circuit 
having  both  resistance  and  inductance,  the  current  does 
not  instantly  assume  its  full  ultimate  value,  but  logarith- 
mically increases  to  that  value. 

At  the  instant  of  closing  the  circuit  there  is  no  current 
flowing.  Let  time  be  reckoned  from  this  instant.  At 
any  subsequent  instant,  t  seconds  later,  the  impressed 
E.M.F.  may  be  considered  as  the  sum  of  two  parts,  El 
and  Er.  The  first,  Elt  is  that  part  which  is  opposed  to, 
and  just  neutralizes,  the  E.M.F.  of  self-induction,  so  that 


but 


The  second  part,  Er)  is  that  which  is  necessary  to  send 
current  through  the  resistance  of  the  circuit,  according  to 
Ohm's  Law,  so  that 

Er  =  RL 

If  the  impressed  E.M.F. 


then  (E  -  RI)  dt  =  Ldl, 

L  L        Rdl 

and  dt  =  -=  -  =•=  ///=  —  = 


E  -  RI  R    E-  RI 


SELF-INDUCTION. 


21 


Integrating  from   the  initial   conditions  /  =  o,  7=o  to  any 
conditions  t  —  t,  /=/', 

-  ^  Pog  (E  -  RI')  -  \og£] 
Rt 


and 


where  e  is  the  base  of  the  natural  system  of  logarithms. 

This  equation  shows  that  the  rise  of  current  in  such  a 
circuit  is  along  a  logarithmic  curve,  as  stated,  and  that  when 
/  is  of  sufficient  magnitude  to 

'—* 

render  the  term  e  L  negli- 
gible the  current  will  follow 
Ohm's  Law,  a  condition  that 
agrees  with  observed  facts. 

/  Fig.  1 5  shows  the  curve  of 
growth  of  current  in  the  coil 
referred  to  in  §  9.     The  curve 
is    calculated   by   the    above   formula    for    the    conditions 
noted. 

The  ratio  —  is  called  the  time  constant  of  the  circuit, 
R 

for  the  greater  this  ratio  is,  the  longer  it  takes  the  current 
to  obtain  its  full  ultimate  value. 


0    .01    ,02   .03    . 04  . 05    .08    . 07 
SECONDS 

Fig.  15- 


12.  Decay  of  Current  in  an  Inductive  Circuit.  —  If  a  cur- 
rent be  flowing  in  a  circuit  containing  inductance  and  re- 
sistance, and  the  supply  of  E.M.P\  be  discontinued, 
without,  however,  interrupting  the  continuity  of  the  circuit, 
the  current  will  not  cease  instantly,  but  the  E.M.F.  of 


22  ALTERNATING-CURRENT    MACHINES. 

self-induction  will  keep  it  flowing  for  a  time,  with  values 
decreasing  according  to  a  logarithmic  law. 

An  expression  for  the  value  of  this  current  at  any  time, 
t  seconds  after  cutting  off  the  source  of  impressed 
E.M.F.,  may  be  obtained  as  in  the  preceding  section.  Let 
time  be  reckoned  from  the  instant  of  interruption  of  the 
impressed  E.M.F.  The  current  at  this  instant  may  be 

represented  by  — ,  and  is  due  solely  to  the  E.M.F.  of  self- 
induction. 

Therefore,  from  Ohm's  Law, 


Integrating  from   the  initial  conditions  t  =  o,  /  =  —  -,  to 
the  conditions,  /  =  /,  7  =  /', 

j°  r ..      L  rf'f< 

9 

DECAYING  CURRENT 

E.M.F.- 0  R 

L  ,       /' 


2 

and      7'=f-e-z', 
yt 

which  is  seen  to  be  the  term  that  had  to  be  subtracted  in 
the  formula  for  growth  of  current.  This  shows  clearly 
that  while  self-induction  prevents  the  instantaneous  attain- 
ment of  the  normal  value  of  current,  there  is  eventually  no 
loss  of  energy,  since  what  is  subtracted  from  the  growing 
current  is  giver  back  to  the  decaying  current. 

Fig.  1 6  is  the  curve  of  decay  of  current  in  the  same  cir- 


SELF-INDUCTION.  23 

cuit  as  was  considered  in  Fig.  15.  The  ordinates  of  the 
one  figure  are  seen  to  be  complementary  to  those  of 
the  other. 

13.  Magnetic  Energy  of  a  Started  Current.  —  If  a  cur- 
rent /  is  flowing  under  the  pressure  of  E  volts,  the  power 
expenditure  is  El  watts,  and  the  work  performed  in  the 
interval  of  time  dt  is 

dW  =  Eldt. 

But  in  a  coil  of  ;/  turns,  the  E.M.F.  induced  by  a  change 

of  linkages  is 

_ 

~ 
Substituting, 

dW^-—-d®. 

io8 

If  the  -circuit  have  a  constant  permeability, 


=  ldi=  \< 
so  dW=-LIdL 


Integrating  through  the  full   range,  from  o  to  Wand  from 
o  to  /, 

f*W  f*I 

I    dW  =  -  L   I  Idl, 
«7o  J  o 

W=- 


Which  is  an  expression  for  the  work  done  upon  the  mag- 
netic field  in  starting  the  current.  When  the  current  is 
stopped  the  work  is  done  by  the  field,  and  the  energy  is 
returned  to  the  circuit. 

This  formula  assumes  the  value  of  L  to  be  constant  dur- 
ing the  rise  and  fall  of  the  current.  If  there  be  iron  in 
the  magnetic  circuit  the  relation  nd$  =  Idi  becomes  nd& 


24  ALTERNATING-CURRENT    MACHINES. 


=  I'dij  I'  being  also  a  variable  ;  but  an  average  of  the  val- 
ues of  /'  throughout  the  range  may  be  called  /,  and  the 
formula  for  energy  stored  in  the  field  holds  true. 

Since  iron  has  always  a  hysteretic  loss,  some  of  the 
energy  is  consumed,  and  the  work  given  back  at  the  dis- 
appearance of  the  field  is  less  than  that  used  to  establish 
the  field  by  the  amount  consumed  in  hysteresis. 

14.   Current  Produced  by  a  Harmonic  E.M.F.  in  a  Cir- 
cuit Having  Resistance  and  Inductance.  —  Given  a  circuit 
of  resistance  R  and  inductance  L  upon  which  is  impressed. 
a  harmonic  E.M.F.  E  of  frequency  /,  to  find  the  current 
/  in  that  circuit. 

Represent  by  w  the  quantity  2w/V 

At  any  instant  of  time,  /,  let  the  instantaneous  value  of 
the  current  be  /'. 

To  maintain  this  current  requires  an  E.M.F.  whose  value 
at  this  instant  is  I'  R.  Represent  this  by  E'r. 

From  §  3,  in  a  harmonic  current, 

fr=fm  sin  to/, 
hence,  -#/=  £fm  s'm  w/t- 

Evidently  Erf  has  its  maximum  value  l?fm  =  Erm  at  (0^  =  90° 
or  270°,  and  its  effective  value  is  Er  —  RI. 

The  counter  E.M.F.  of  self-induction  at  the  same  instant 
of  time,  /,  is 

|;  *•--*£• 

But  as  before,  /'=/«,  sin  <o/, 

so  dl'  =  <o/m  cos  at  dt, 

and  J£{  =  —  ^LIm  cos  o>£ 


SELF-INDUCTION.  2$ 

Evidently  £s'  has  a  maximum  value  of   —  &LIm  =  Esm  at 
iot  =  o°  or  1  80°,  and  its  effective  value 

Es  =  -  <oZ7. 

It  is  clear  that  the  impressed  E.M.F.  must  be  of  such 
a  value  as  to  neutralize  Es  and  also  supply  Er.  But  these 
two  pressures  cannot  be  simply 
added,  since  the  maximum  value  of 
one  occurs  at  the  zero  value  of  the 
other;  that  is,  they  are  at  right 
angles  to  each  other,  as  defined  in  m 

§   5.      Reference    to    Fig.    17    will  Fig'  I7' 

make   it    clear  that   combining  these  at  right  angles  will 


give  as  a  resultant  the  pressure  V^r2  +  E*  ;  and  it  is  this 
pressure  that  the  impressed  E.M.F.  E  must  equal  and 
oppose.  So 


E  =  V(Z7?)2  +  (<uZ/)2, 

from  which 

j? 

T  


This  is  a  formula  which  must  be  used  in  place  of  Ohm's 
Law  when  treating  inductive  circuits  carrying  harmonic 
currents.  It  is  evident  that,  if  the  inductance  or  the  fre- 
quency be  negligibly  small  (direct  current  has  f  =  o),  the 
formula  reduces  to  Ohm's  Law;  but  for  any  sensible  val- 
ues of  w  and  L  the  current  in  the  circuit  will  be  less  than 
that  called  for  by  Ohm's  Law. 

The  expression  V^2  +  o>2Z2  is  called  the  impedance  of 
the  circuit,  and  also  the  apparent  resistance.  The  term  R 
is  of  course  called  resistance,  while  the  term  o>Z,  which  is 
2  irfL,  is  called  the  reactance.  Both  are  measured  in  ohms. 

The  effective  value  of  the  counter  E.M.F.  of  self-indue- 


26  ALTERNATING-CURRENT    MACHINES. 

tion  can  be  determined  as  follows,  without  employing  the 
calculus  ;  that  it  must  be  combined  at  right  angles  with 
RI  is  not  directly  evident.  Disregarding  the  direction  of 
flow,  an  alternating  current  i  reaches  a  maximum  value  im 
2f  times  per  second.  The  maximum  number  of  lines  of 
force  linked  with  the  circuit  on  each  of  these  occasions  is 
lim.  The  interval  of  time,  from  when  the  current  is  zero 
with  no  linkages,  to  when  the  current  is  a  maximum  with 

lim  linkages,  is  —  -  second.      The  average  rate  of  cutting 
4/ 

lines,  then,  is  —  -  ,  and  is  equal  to  the  average  E.M.F.  of 

4? 

self-induction  during  the  interval.     It  has  the  same  value 

during  succeeding  equal  intervals  ;  i.e., 


4? 

The  effective  value  is  (§  4)  therefore, 

es  =  —  2  Trfli  =  uli, 
and  in  practical  units, 

E.  =  -  2  7T/Z7. 

Since  the  squares  of  the  quantities  R,  L,  and  o>  enter 
into  the  expression  for.  the  impedance,  if  one,  say  R,  is 
moderately  small  when  compared  with  L  or  to,  its  square 
will  be  negligibly  small  when  compared  with  L?  or  <o2.  The 
frequency,  because  it  is  a  part  of  cu,  may  be  a  considerable 
factor  in  determining  the  impedance  of  a  circuit. 

Having  recourse  once  again  to  the  harmonic  shadow- 
graph described  in  §  3,  the  phase  relation  between  im- 
pressed E.M.F.  and  current  may  be  made  plain.  It  has 
already  been  shown  that  Er  and  EB  are  at  right  angles  to 


SELF-INDUCTION.  27 

each  other.     Since  the  pressure  Er  is  the  part  of  the  im-' 
pressed  E.M.F.  which  sends  the  current,  the  current  must 
be  in  phase  with  it.     Therefore  there  is  always  a  phase 
displacement  of  90°  between  /  and  Es.     This  relation  is 
also  evident  from  a  consideration  of  the  fact  that  when  / 
reaches  its  maximum  value  it  has,  for  the  instant,  no  rate 
of  change  ;  hence  the  flux,  which  is  in  phase  with  the  cur- 
rent, is  not  changing,  and  consequently  the  E.M.F.  of  self- 
induction  must  be,  for  the  instant,  zero.     That  is,  /  is  maxi- 
mum when  Es  is  zero,  which  means  a  displacement  of  90°. 
In  Fig.  1 8  the  triangle  of  E.M.F.'s  of  Fig.  17  is  altered 
to  the  corresponding  parallelogram  of  E.M.F. 's,  and  the 
maximum  values  substituted  for  the  effective.      If  now  the 
parallelogram  re- 
volve   about    the 
center      o,      the 
traces  of  the  har- 
monic shadows  of 
the  extremities  of 
Em,   Erm  and  Esm 
will     develop    as 
shown.      It  is  evident  that  the  curve  Er'  —  and  so  also  the 
curve  of  current  —  lags  behind  the  curve  E'  by  the  angle 
<f>.     It  is  clear  that  the  magnitude  of  <f>  depends  upon  the 
relative  values  of  L  and  R  in  the  circuit,  the  exact  relation 
being  derived  from  the  triangle  of  forces. 


36CT 


Fig.  18. 


Es  <oZ/  <uZ  2  7T/Z 


Furthermore 


that  is,  the  cosine  of  the  angle  of  lag  is  equal  to  the  ratio 


28  ALTERNATING-CURRENT    MACHINES. 

of  the  volts  actually  engaged  in  sending  current  to  the 
volts  impressed  in  the  circuit,  and  this  ratio  is  again  equal 
to  the  power-factor  as  stated  in  §  6. 

15.  Choke  Coils.  —  The  term  choke  coil  is  applied  to 
any  device  designed  to  utilize  counter  electromotive  force 
of  self-induction  to  cut  down  the  flow  of  current  in  an 
alternating-current  circuit.  Disregarding  losses  by  hyster- 
esis, a  choke  coil  does  not  absorb  any  power,  except  that 
which  is  due  to  the  current  passing  through  its  resistance. 
It  can  therefore  be  more  economically  used  than  a  rheostat 
which  would  perform  the  same  functions. 

These  coils  are  often  used  on  alternating-current  cir- 
cuits in  such  places  as  resistances  are  used  on  direct- 
current  circuits.  For  instance,  in  the  starting  devices 
employed  in  connection  with  alternating-current  motors, 
the  counter  E.M.F.  of  inductance  is  made  to  cut  down  the 
pressure  applied  at  the  motor  terminals.  The  starter  for 
direct-current  motors  employs  resistance. 

Since  a  lightning  discharge  is  oscillatory  in  character 
and  of  enormous  frequency,  a  coil  which  would  offer  a 
negligible  impedance  to  an  ordinary  alternating  current 
will  offer  a  high  impedance  to  a  lightning  discharge.  This 
fact  is  recognized  in  the  construction  of  lightning  arresters. 
A  choke  coil  of  but  few  turns  will  offer  so  great  an  impe- 
dance to  a  lightning  discharge  that  the  high-tension,  high- 
frequency  current  will  find  an  easier  path  to  the  ground 
through  an  air  gap  suitably  provided  than  through  the 
machinery,  and  the  latter  is  thus  protected. 

Choke  coils  are  also  used  in  connection  with  alternating- 
current  incandescent  lamps,  to  vary  the  current  passing 
through  them,  and  in  consequence  to  vary  the  brilliancy.  , 


CAPACITY.  29 


CHAPTER    III. 

CAPACITY. 

16.  Condensers. — Any  two  conductors  separated  by  a 
dielectric  constitute  a  condenser.  In  practice  the  word  is 
generally  applied  to  a  collection  of  thin  sheets  of  metal 
separated  by  thin  sheets  of  dielectric,  every  alternate 
metal  plate  being  connected  to  one  terminal  of  the  instru- 
ment, the  intervening  plates  to  the  other  terminal.  The 
Ley  den  jar  is  also  a  common  form  of  condenser. 

The  office  of  a  condenser  is  to  store  electrical  energy  by 
utilizing  the  principle  of  electrostatic  induction.  If  a  con- 
tinuous E.M.F.  be  applied  to  the  terminals  of  a  condenser, 
a  current  will  flow,  large  at  first  and  gradually  diminish- 
ing, till  the  plates  of  the  condenser  have  been  charged  to 
an  electrostatic  difference  of  potential  that  equals  and 
opposes  the  electrodynamic  pressure  applied.  Then  there 
is  a  balance  of  E.M.F.'s,  and  no  current  will  flow  if  there 
be  no  leakage. 

A  frequent  misconception  as  to  the  capacity  of  a  con- 
denser is  that  it  is  equal  to  the  quantity  of  electricity  it 
will  hold.  The  quantity  of  electricity  a  given  condenser 
will  hold  is  directly  proportional  to  the  tension  of  the 
charge,  and  a  consideration  of  this  fact  leads  to  the  follow- 
ing definition  : 

The  capacity  of  a  condenser  is  numerically  equal  to  the 
quantity  of  electricity  with  which  it  must  be  charged  in 


30  ALTERNATING-CURRENT    MACHINES. 

order  to  raise  the  potential  difference  between  its  terminals 
from  zero  to  unity. 

If  the  quantity  and  potential  be  measured  in  c.  G.  s. 
units,  the  capacity,  cy  will  be  in  c.  G.  s.  units.  If  practical 
units  be  employed,  the  capacity,  c,  is  expressed  in  farads. 
The  farad  is  the  practical  unit  of  capacity.  A  condenser 
whose  potential  is  raised  one  volt  by  a  charge  of  one  cou- 
lomb has  one  farad  capacity.  The  farad  is  io~9  times  the 
absolute  unit,  and  even  then  is  too  large  to  conveniently 
express  the  magnitudes  encountered  in  practice.  The 
term  microfarad  (y-oo"-0"oo"o  faracO  is  m  most  general 
use. 

In  electrostatics,  both  air  and  glass  are  used  as  dielec- 
trics in  condensers  ;  but  the  mechanical  difficulties  of  con- 
struction necessitate  a  low  capacity  per  unit  volume,  and 
therefore  render  these  substances  impracticable  in  electro- 
dynamic  engineering.  Mica,  although  it  is  expensive  and 
difficult  of  manipulation,  is  generally  used  as  the  dielectric 
in  standard  condensers  and  in  those  which  are  intended  to 
withstand  high  voltages.  Many  commercial  condensers 
are  made  from  sheets  of  tinfoil,  alternating  with  slightly 
larger  sheets  of  paraffined  paper.  Though  not  so  good  as 
mica,  paraffin  will  make  a  good  dielectric  if  properly 
treated.  It  is  essential  that  all  the  moisture  be  expelled 
from  the  paraffin  when  employed  in  a  condenser.  If  it  is 
not,  the  water  particles  are  alternately  attracted  and 
repelled  by  the  changes  of  potential  on  the  contiguous 
plates,  till,  by  a  purely  mechanical  action,  a  hole  is  worn 
completely  through  the  dielectric,  and  the  whole  condenser 
rendered  useless  by  short-circuit.  Ordinary  paper  almost 
invariably  contains  small  particles  of  metal,  which  become 
detached  from  the  calendar  rolls  used  in  manufacture. 


OF  THE  ^ 

DIVERSITY  1  < 

OF  / 

CAPACITY.  31 

These  occasion  short-circuits  even  when  the  paper  is 
doubled. 

A  distinctly  different  form  of  condenser  is  the  electro- 
lytic condenser.  It  consists  of  two  electrodes  dipping 
into  an  electrolyte,  as,  for  instance,  two  carbon  electrodes 
in  zinc  sulphate.  A  charge  of  electricity  will  deposit 
zinc  upon  one  electrode  and  set  up  an  E.M.F.  of  polariza- 
tion. Such  condensers  should  not  be  subjected  to  volt- 
ages in  excess  of  their  E.M.F.  of  polarization.  Electro- 
lytic condensers  have  about  the  same  volume  as  other 
condensers  of  the  same  volt-ampere  capacity. 

The  maximum  voltage  that  may  be  applied  to  a  con- 
denser is  limited  by  the  dielectric  strength  of  the  material 
employed.  If  this  limit  be  exceeded,  the  dielectric  will 
be  ruptured,  which  renders  the  condenser  useless.  The 
ohmic  resistance  of  condenser  dielectrics  is  not  infinite. 
There  is  always  a  leakage  from  one  charged  plate  to  the 
other  through  the  insulation  and  over  its  surface.  Poor 
insulation  may  occasion  a  considerable  loss,  which  appears 
as  heat  in  the  apparatus  when  in  use.  There  is  also  a 
dielectric  hysteresis  which  is  analogous  to  magnetic  hystere- 
sis in  iron.  A  dielectric  with  a  high  hysteretic  constant 
may  consume  considerable  power  when  in  use,  which  will 
also  appear  as  heat. 

The  capacity  of  a  condenser  may  be  calculated  by  using 
the  following  formula  :  — 


C  =  .000225  An 
T    ' 


where 


C  —  capacity  in  microfarads, 

A  =  area  of  dielectric  between  two  conducting  plates 
in  square  inches, 


ALTERNATING-CURRENT    MACHINES. 


n  =  number  of  sheets  of  dielectric, 
/  =  thickness  of  dielectric  in  mils, 

k  =  specific  inductive  capacity  of    dielectric  as  obtained 
from  the  following  table. 

TABLE. 

Glass 3     to  7 

Ebonite 2.2  to  3 

Gutta-percha 2.5 

Paraffin 2      to  2.3 

Shellac 2.75 

Mica 6.6 

Beeswax 1.8 

Kerosene 2      to  2.5 

17.   Connection  of  Condensers  in  Parallel  and  in  Series. 

-Condensers  may  be  connected  in  parallel  as  in  Fig.  19. 
If    the  capacities  of    the    individual 
condensers  be  respectively  Clf  C9,  C8, 
etc.,  the  capacity  C  of  the  combina- 
tion  will  be 


The  parallel  arrangement  of  sev- 
Flg<  I9*  eral  condensers  is  equivalent  to  in- 

creasing the  number  of  plates  in  one  condenser.  An 
increase  in  the  number  of  plates  results  in  an  increase  in 
the  quantity  of  electricity  necessary  to  raise  the  potential 
difference  between  the  terminals  of  the  condenser  one 
volt ;  that  is,  an  increase  in  the  capacity  results. 

If  the  condensers  be  connected  in  series,  as  in  Fig.  20, 
the  capacity  of  the  combination  will  be 

C  = 


CAPACITY.  33 

For,  if  a  quantity  of  positive  electricity,  Q,  flow  into  the 

left  side  of  C^  it  will  induce  and  keep  bound  an  equal  neg- 

ative quantity  on  the  right  side  of  C^  and  will  repel  an 

equal  positive  quantity.     This  last  quantity  will  constitute 

the  charge  for  the 

left    side     of    C2. 

The    operation    is 

repeated     in    the 

case    of    each    of  Fi«-  20- 

the  condensers.     It    is    thus    clear    that    the  quantity   of 

charge  in  each  condenser  is   Q.     The  impressed  E.M.F. 

must  consist  of  the  sum  of  the  potential  differences  on  the 

separate  condensers.     Let  these  differences  be  respectively 

Elt  Ep  E#  etc.     Then  the  impressed  E.M.F. 


But  £,  =  E,  =  £,=     >    etc., 

C-1  C2  U3 

and  also,  E  —  ^, 

C- 

therefore  =       +      =      +  •••• 


or 


'2  ^3 

I 


As  an  example,  consider  three  condensers  of  respective 
capacities  of  i,  2,  and  5  microfarads.  Since  the  factor  to 
reduce  to  farads  will  appear  on  both  sides  of  the  equations, 
it  may  here  be  omitted.  With  the  three  in  multiple  (Fig. 
19),  the  capacity  of  the  combination  will  be 

C  =  i  +  2  +  5  =  8  mf . 


34  ALTERNATING-CURRENT    MACHINES. 

With  the  three  in  series  (Fig.  20), 

i 


C  = 


=  .588  mf. 


With  the  two  smaller  in  parallel  and  in  series  with  the 
larger  (Fig.  21), 

C=-  -=  1.875  mf. 

1  +  2  +5 


Fig.  21. 

With  the  two  smaller  in  series  and  in  parallel  with  the 
larger  (Fig.  22), 


C  = 


+  5  =  5-666  mf. 


If  with  any  condensers 

Q  =  Ci 
then,  with  H  in  multiple, 

and  with  n  in  series, 


Cn, 


C  =  nClt 


C  =  -  C,. 


It  is  interesting  to  note  that  the  formulas  for  capacities 
in  parallel  and  in  series  respectively  are  just  the  reverse  of 
those  for  resistances  in  parallel  and  in  series  respectively. 

18.  Growth  of  Current  in  a  Condensive  Circuit.  —  The 
opposition  to  a  flow  of  current  which  is  caused  by  a  con- 


CAPACITY.  35 

denser  is  quite  different  from  that  which  is  caused  by  a 
resistance.  To  be  sure,  there  is  some  resistance  in  the 
leads  and  condenser  plates,  but  this  is  generally  so  small 
as  to  be  negligible.  The  practically  infinite  resistance  of 
the  condenser  dielectric  does  not  obstruct  the  current  as 
an  ordinary  resistance  is  generally  considered  to  do.  The 
dielectric  is  the  seat  of  a  polarization  E.M.F.  which  is  de- 
veloped by  the  condenser  charge  and  which  grows  with  it. 
It  is  a  counter  E.M.F.  ;  and  when  it  reaches  a  value  equal 
to  that  of  the  impressed  voltage,  the  charging  current  is 
forced  to  cease. 

To  find  the  current  at  any  instant  of  time,  /",  in  a  circuit 
(Fig.  23)  containing  a  resistance  R  and  a  capacity  C,  the 
constant  impressed  pressure  E  must 
be  considered  as   consisting  of  two 
variable  parts,  one  Er)  being  active 
in  sending  current  through  the  re- 
sistance, and    the    other    part,  Ec, 
being   required    to    balance  the    po- 
tential of  the  condenser.     Then  at 
all   times 

Let  time  be  reckoned  from  the  instant  the  pressure  E  is 
applied  ;  when,  therefore,  /  =  o  and  /0  =  —.     Consider  the 

current  at  any  instant  of  time  to  be  /'.  Then  if  it  flow 
for  dt  seconds  it  will  cause  dQ  coulombs  to  traverse  the 
circuit,  and 

r/_    <*Q 


or 
at 


from  which 


<r=//v, 


36  ALTERNATING-CURRENT    MACHINES. 

Qf 


By  definition, 

therefore, 

E'— 

• 

And  by  Ohm's  Law, 

so  at  this  instant  of  time 


r 
J 


c 


whence  EC  = 

which  upon  differentiating,  becomes 
Integrating 


C*  ,f  =  _  RC  T  ^ 

Jo  J/o       / 


CONDENSER 
CHARGING  CURRENT 
E— 100  V. 
R-10 
C  — 2  MF. 


Solving  for  /', 


which  is  the  expression 
sought.  Like  the  corre- 
sponding expression  for  an 
inductive  circuit,  it  is  loga- 
rithmic. 

Fig.  24  is  a  curve  showing  the  growth  of  current   in  a 
condenser  for  the  conditions  indicated. 


CAPACITY.  37 

19.  Condensers  in  Alternating-Current  Circuits  —  Hy- 
draulic Analogy.  —  Imagine  a  circuit  consisting  of  a  pipe 
through  which  water  is  made  to  flow,  first  one  way,  then 
the  other,  by  a  piston  oscillated  pump-like  in  one  section 
of  it.  The  pipe  circuit  corresponds  to  an  electric  circuit, 
the  pump  to  a  generator  of  alternating  E.M.F.,  and  the 
flow  of  water  to  a  flow  of  alternating  current.  Further 
imagine  one  section  of  the  pipe  to  be  enlarged,  and  in  it 
placed  a  transverse  elastic  diaphragm.  This  section  cor- 
responds to  a  condenser.  Its  capacity  with  a  unit  pressure 
of  water  on  one  side  depends  upon  the  area  of  the  dia- 
phragm, its  thinness,  and  the  elastic  co-efficients  of  the  ma- 
terial of  which  it  is  made.  In  a  condenser  the  capacity 
depends  upon  the  area  of  the  dielectric  under  strain,  its 
thinness,  and  the  specific  inductive  capacity  of  the  dielec- 
tric employed.  As  the  water  surges  to  and  fro  in  the 
pipe,  some  work  must  be  done  upon  the  diaphragm,  since. 
it  is  not  perfectly  elastic.  This  loss  corresponds  to  the 
loss  in  a  condenser  by  dielectric  hysteresis.  The  fact  that 
the  diaphragm  is  not  absolutely  impervious  to  water  cor- 
responds to  the  fact  that  a  dielectric  is  not  an  absolute 
electric  insulator.  As  the  diaphragm  may  be  burst  by  too 
great  a  hydrostatic  pressure,,  so  may  the  dielectric  be  rup- 
tured by  too  great 


20.  Phase  Relations.  —  To  understand  the  relation  be- 
tween pressure  and  current  in  a  condensive  circuit,  con- 
sider the  above  analogy.  Imagine  the  diaphragm  in  its 
medial  position,  with  equal  volumes  of  water  on  either  side 
of  it,  and  the  piston  in  the  middle  of  its  travel.  This 
middle  point  corresponds  to  zero  pressure.  When  the  pis- 
ton is  completely  depressed,  there  is  a  maximum  negative 


38  ALTERNATING-CURRENT   MACHINES. 

pressure,  when  completely  elevated,  a  maximum  positive 
pressure,  if  pressure  and  flow  upward  be  considered  in  the 
positive  direction.  If  the  piston  oscillate  in  its  path  with  a 
regular  motion,  it  is  clear  that  the  water  will  flow  upward 
from  the  extreme  lowest  to  the  extreme  highest  position  of 
the  piston.  That  is,  there  will  be  flow  in  the  positive  direc- 
tion from  the  maximum  negative  to  the  maximum  positive 
values  of  pressure.  The  direction  of  flow  is  seen  to 
remain  unchanged  while  the  piston  passes  through  its 
middle  position  or  the  point  of  zero  pressure. 

Returning  to  electric  phenomena,  if  a  harmonic  E.M.F. 
be  impressed  upon  any  circuit,  a  harmonic  current  will 

flow  in  it.  So  in  a  cir- 
cuit containing  a  con- 
denser and  subject  to  a 
sinusoidal  E.M.F.,  the 
current  flow  will  be  sinu- 
soidal. This  flow  will  be 
Flg'  25'  in  the  positive  direction 

from  the  negative  maximum  to  the  positive  maximum  of 
pressure,  and  in  a  negative  direction  from  the  positive 
maximum  to  the  negative  maximum,  as  described  above. 
This  necessitates  that  the  zero  values  of  current  occur  at 
the  maximum  values  of  pressure  ;  and  since  the  curves  are 
both  sinusoids,  their  relation  may  be  plotted  as  in  Fig.  25. 
It  is  immediately  seen  that  these  curves  are  at  right 
angles,  as  described  in  §  5,  and  that  the  current  leads  the 
pressure  by  90°. 

Reference  again  to  the  hydraulic  analogy  will  show  that 
the  condenser  is  completely  charged  at  the  instant  of 
maximum  positive  pressure,  discharged  at  the  instant  of 
zero  pressure,  charged  in  the  opposite  direction  at  the  in- 


CAPACITY. 


39 


Fig.  26. 


slant  of  maximum  negative 

pressure,    and    finally    dis- 

charged at  the  instant    of 

the     next    zero    pressure. 

Thus  the  charge  is  zero  at 

the  maximum  current  flow, 

and  at  a  maximum  at  zero 

current,  that  is,  when  the 

current  turns  and  starts  to  flow  out.      These  points  are 

marked  in  Fig.   26. 

21.    Current    and   Voltage    Relations.  —  If  a  sinusoidal 
pressure  E  of  frequency/"  be  impressed  upon  a  condenser, 

the  latter  is   charged  in  -  .  —  seconds,  discharged  in  the 

i  4 

next  —  seconds,  and  charged  and  discharged  in  the  oppo- 

4/ 

site  direction  in.  the  equal  succeeding  intervals.  The 
maximum  voltage  Em  =  ^E  (§  4),  hence  the  quantity  at 
full  charge  is 


The  quantity  flowing  through  the  circuit  per  second  is 


This  number  therefore  represents  the  average  current,  or 

'  Iav  =  4 
From  §  4,  the  effective  current 


whence 
and 


2    V2 
1=   2 

I 


E  = 


27T/C     ' 


ALTERNATING-CURRENT    MACHINES. 


The  last  is  an  expression  for  the  volts  necessary  to  send 
the  capacity  current  through  a  circuit.  The  expression 

— ^is  called  the  capacity  reactance  of  the  circuit.     It  is 

analogous  to  2  -nfL,  the  inductance  reactance  of  an  induc- 
tive circuit. 

If  the  circuit  contain  both  a  resistance  R  and  a  capa- 
city C,  the  voltage  E  impressed  upon  it  must  be  considered 
as  made  up  of  two  parts,  Er,  which  sends  current  through 
the  resistance  and  is  therefore  in  phase  with  the  current, 
and  Ecy  which  balances  the  counter  pressure  of  the  conden- 
ser and  is  therefore  90°  behind  the  current  in  phase. 

By  Ohm's  Law 

T?  ~D  T 

£,r  =  Kl, 

and  from  above 

/ 


27T/C 

The  impressed  E  must  overcome  the   resultant  of  these 
two  E.M.F.'s ;  and  since  they  are  at  right  angles 


Er~RI 


or 

o 

p 


Fig.  37. 

The  relation  of  the  E.M.F.'s  is  shown  graphically  in 
Fig.  27,  where  the  current,  which  is  in  phase  with  the 
pressure  Er,  is  seen  to  lead  the  impressed  pressure  by  the 
angle  <£. 

22.  Resistance,  Inductance,  and  Capacity  in  an  Alter- 
nating-Current Circuit.  —  The  general  case  of  an  alter- 


CAPACITY. 


nat  ing-cur  rent  circuit  is  one  that  contains  resistance, 
inductance,  and  capacity.  To  derive  the  expression  for 
current  flow  in  such  a  circuit,  it  is  but  necessary  to  com- 
bine the  results  already  found ;  and  this  is  most  readily 
done  graphically.  In  §  14  it  was  shown  that  the  counter 


27T/L 


27T/L 


Fig,  28. 

E.M.F.  due  to  the  inductance  reactance   of  a  circuit  is 
2-nfL,  and  leads  the  current  by  90°.     In  §21  it  was  shown 

that  the  E.M.F.  of  capacity  reactance  of  a  circuit  is -^=. 

2   7T/C 

and  lags  behind  the  current  by  90°.     These  two  E.M.F.'s 
are,  then,  in  exactly  opposite  phases,  or   180°  apart,  and 


Fig.  29. 

the  resultant  reactance  is  merely  their  numerical  difference. 
These  relations  are  shown  in  Fig.  28,  where  the  reactance 
of  inductance  is  greater  than  that  of  condensance,  and  in 
Fig.  29,  where  the  latter  exceeds  the  former,  the  resistance 
being  the  same  in  either  case.  Clearly  the  impedance 
resulting  from  the  three  factors  R,  L,  and  C  is  represented 


42  ALTERNATING-CURRENT    MACHINES. 

ill  direction  and  in  magnitude  by  the  hypothenuse  as  shown, 
and  the  impressed  pressure  is  proportional  to  this  quantity. 
The  general   expression  for  the  flow   of  an  alternating 
current  through  any  kind  of  circuit  is  therefore 


the  quantity  within  the  brackets  indicating  an  angle  of  lag 
of  current  if  positive,  and  an  angle  of  lead  if  negative. 

23.    Resonance.  —  If  in  a  circuit  containing  inductance 
and  capacity  as  well  as  resistance,   the   two  former  are 


proportioned  so  that 
the  expression 


reduces  to 


the  capacity  being  of  a  proper  magnitude  to  balance  induc- 
tance. At  one  instant  energy  is  being  stored  in  the  field 
at  the  same  rate  as  it  is  being  given  to  the  circuit  by  the 
condenser,  and  at  another  instant  energy  is  being  released 
from  the  field  at  the  same  rate  as  it  is  being  stored  in'  the 
condenser. 

When  this  condition  prevails,  resonance  is  said  to  be 
attained,  or  the  circuit  is  said  to  be  in  tune. 

If  the  capacity  and  inductance  be  in  parallel,  enormous 


CAPACITY.  43 

currents  may  flow  between  the  two.  This  is  because  the 
two  are  balanced,  and  the  one  is  at  any  time  ready  to 
receive  the  energy  given  up  by  the  other ;  and  a  surging 
once  started  between  them  receives  periodical  increments 
of  energy  from  the  line.  This  is  analogous  to  the  well- 
known  mechanical  phenomena  that  a  number  of  gentle, 
but  well-timed,  mechanical  impulses  can  set  a  very  heavy 
suspended  body  into  violent  motion.  The  frequency  of 
these  impulses  must  correspond  exactly  to  the  natural 
period  of  oscillation  of  the  body. 

If  the  capacity  and  inductance  be  in  series,  the  differ- 
ence of  potential  between  the  terminals  of  either  may  be 
far  greater  than  the  E.M.F.  impressed  upon  the  circuit. 

In  the  first  case  damage  is  likely  to  result  from  the 
overloading  of  the  conductors  between  the  inductance  and 
the  capacity,  even  to  burning  them  out,  while  in  the  second 
case  the  pressure  may  rise  to  such  a  point  as  to  puncture 
the  insulation  of  all  the  apparatus  in  the  circuit,  even  that 
of  the  generator  itself. 


44  ALTERNATING-CURRENT   MACHINES. 


CHAPTER   IV. 

PROBLEMS  ON  ALTERNATING-CURRENT 
CIRCUITS. 

24.  Definitions  of  Terms.  —  In  considering  the  flow  of 
alternating  currents  through  series  circuits  and  through 
parallel  circuits,  continual  use  must  be  made  of  various 
expressions,  some  of  which  have  been  defined  during  the 
development  of  the  previous  chapters.  For  convenience 
the  names  of  all  the  expressions  connected  with  the  general 
equation 

E 


v 


7T/Z 


-M 

**fc) 


will  be  given  and  defined. 

/  is  the  current  flowing  in  the  circuit.  It  is  expressed 
in  amperes,  and  lags  behind  or  leads  the  pressure,  by  an 
angle  whose  value  is 


=  tan 


E  is  the  harmonic  pressure,  of  maximum  value  %/2  E, 
which  is  applied  to  the  circuit,  and  has  a  frequency/.  It 
is  expressed  in  volts. 

R  is  the  resistance  of  the  circuit,  and  is  expressed  in 
ohms.  It  is  numerically  equal  to  the  product  of  the  im- 
pedance by  the  cosine  of  <£. 


PROBLEMS. 


45 


L  is  the  inductance  of  the  circuit,  and  is  expressed  in 
henrys. 

C  is  the  localized  capacity  of  the  circuit,  and  is  expressed 
in  farads. 

2  TrfL  is  the  inductive  reactance  of  the  circuit,  and  is 
expressed  in  ohms. 


27T/C 

is  the  capacity  reactance,  or  capacitance,  of  the  circuit,  and 
is  expressed  in  ohms. 


is  the  reactance  of  the  circuit,  and  is  expressed  in  ohms. 
It  is  numerically  equal  to  the  product  of  the  impedance  by 
the  sine  of  <£. 


is  the  impedance  or  apparent  resistance  of  the  circuit,  and 
is  expressed  in  ohms. 


the  reciprocal  of  the  impedance,  is  the  admittance  of  the 
circuit.     It  is  expressed  in  terms  of  a  unit  that  has  never 


CONDUCTANCE 


Fig.  30. 

been  officially  named,  but  which  has  sometimes  been  called 
the  mho.  There  are  two  components  of  the  admittance, 
as  shown  in  Fig.  30. 


46  ALTERNATING-CURRENT   MACHINES. 

The  conductance  of  a  circuit  is  that  quantity  by  which 
E  must  be  multiplied  to  give  the  component  of  /  parallel 
to  E.  It  is  measured  in  the  same  units  as  the  admittance, 
and  is  numerically  equal  to 

cos  <£ 

impedance ' 
and  also  to 


The  susceptance  of  a  circuit  is  that  quantity  by  which 
E  must  be  multiplied  to  give  the  component  of  /  perpen- 
dicular to  E.  It  is  measured  in  the  same  units  as  the 
admittance,  and  is  numerically  equal  to 

sin  </> 


impedance  ' 
and  also  to 


It  should  be  noticed  that  while  admittance  is  the  recip- 
rocal of  impedance,  conductance  is  not  the  reciprocal  of 
resistance,  nor  is  susceptance  the  reciprocal  of  reactance. 
This  becomes  evident,  upon  considering  numerical  values 
in  connection  with  the  impedance  right-angled  triangle, 
e.g.,  3,  4,  and  5  for  the  sides. 

25.  E.M.F.'s  in  Series.  —  Alternating  E.M.F.'s  that 
may  be  put  in  series  may  differ  in  magnitude,  in  frequency, 
in  phase  relation,  and  in  form  or  shape  of  wave.  Forms 
other  than  that  of  the  sinusoid  need  not  be  discussed. 


PROBLEMS.  47 

E.M.F.'s  of  different   frequencies   in   series  will  give   an 
irregular  wave-form  whose  maximum  values  will  recur  at 


intervals.  The  duration  of 
common  multiple  of  the 
durations  of  the  component 
half  cycles. 

If  two  harmonic  E.M.F.'s 
of  the  same  frequency  and 
phase  be  in  series,  the  re- 
sulting E.M.F.  is  merely 
the  sum  of  the  separate 
E.M.F.'s.  This  condition  is 
shown  in  Fig.  31,  in  which 
the  two  E.M.F.'s  are  plotted 


these   intervals  is  the  least 


Fig.  31. 


together,  and  the  resulting  E.M.F.  plotted  by  making  its 
instantaneous  values  equal  to  the  sum  of  the  correspond- 
ing instantaneous  values  of  the  component  E.M.F.'s.  The 
maximum  of  the  resultant  E.M.F.  is  evidently 


and  since 


and 


E^ 

V2 

V^ 


J£lm 

— -, 

V2 


as  was  stated. 

If  two  E.M.F.'?,  of  the  same  frequency,  but  exactly 
opposite  in  phase,  be  placed  in  series,  it  may  be  similarly 
shown  that  the  resultant  E.M.F.  is  the  numerical  differ- 
ence of  the  component  E.M.F.'s.  This  case  may  occur  in 
the  operation  of  motors. 

The  most  general  case  that  occurs  is  that  of  a  number 
of  alternating  E.M.F.'s  of  the  same  frequency,  but  of 


48 


ALTERNATING-CURRENT    MACHINES. 


different  magnitudes  and  phase  displacements.  The 
changes  in  magnitude  and  phase  and  the  -oh  ise  elation  of 
the  resulting  curve  of  E.M.F.  are  shown  in  Fij.  j2,  where 
recourse  is  had  once  again  to  the  harmonic  shadowgraph. 
But  two  components,  El  and  £"„  are  treated,  whose  phase 
displacement  is  <£r  The  radii  vectors  E}m  and  E^m  are 
laid  off  from  o  with  the  proper  angle  <£,  between  them, 
and  the  shadows  traced  by  their  extremities  are  shown  in 
the  dotted  curves.  The  instantaneous  value  of  the  result- 
ant E.M.F.  is  the  algebraic  sum  of  the  corresponding  in- 


Fig.  3*. 

stantaneous  values  of  the  component  E.M.F.'s,  and  the 
resultant  curve  of  E.M.F.  is  traced  in  the  figure  by  the 
solid  line.  But  this  solid  curve  is  also  the  trace  of  the  ex- 
tremity of  the  line  Em,  which  is  the  vector  sum  (the  result- 
ant of  the  force  polygon)  of  the  component  pressures,  Eim 
and  EW-  This  is  evident  from  the  fact  that  any  instan- 
taneous value  of  the  resultant  pressure  curve  is  the  sum  of 
the  corresponding  instantaneous  values  of  the  component 
curves,  or  (§  3) 

£'=  Elm  sin  o>/  +  EZm  sin  (o>/  +  <fo). 
Again  from  the  force  polygon 

Em  sin  (w/  +  $)  =  Elm  sin  <o/  +  E»m  sin  (W  +  &). 


PROBLEMS. 


49 


Hence  at  any  instant 

E'=  E    sin 


+ 


wherefore  the  extremity  of  the  line  Em  traces  the  curve  of 
resultant  pressure,  <£  being  its  angular  displacement  from 
EL.  If  a  third  component  E.M.F.  is  to  be  added  in  series, 
it  may  be  combined  with  the  resultant  of  the  first  two  in 
an  exactly  similar  manner. 

So  it  may  be  stated  as  a  general  proposition,  that  if  any 
number  of  harmonic  E.M.F.  's,  of  the  same  frequency,  but 
of  various  magnitudes  and 
phase  displacements,  be 
connected  in  series,  the 
resulting  harmonic  E.M.F. 
will  be  given  in  magnitude 
and  phase  by  the  vector  sum  of  the  component  E.M.F.'  s. 
The  analytic  expressions  for  E  and  <f>  may  be  derived  by 
inspection  of  the  diagram,  and  are 


—  »~-*r—  "—  e-. 


E  = 
and 


]Y 


sn 


Ez  sin  <£2  4- 


cos 


E  cos 


Fig.  34- 


As  a  numerical  example,  suppose  three  alternators,  Fig. 
33,  to  be  connected  in  series.  Suppose  these  to  give  sine 
waves  of  pressure  of  values  El  =  70,  E^  =  6°>  anc^  -^3  =  4° 


ALTERNATING-CURRENT   MACHINES. 


volts  respectively.  Considering  the  phase  of  El  to  be 
the  datum  phase,  let  the  phase  displacements  be  <^i  =  o°, 
<£2  =  4O°,  and  ^  =  75°,  respectively.  It  is  required  to  find 
E  and  <£.  Completing  the  parallelograms  or  completing 
the  force  polygon  as  shown  in  Fig.  34,  it  is  found  that 
E  =  148.7  volts  and  <£  =  32.1°. 

26.  Polygon  of  Impedances. — Consider  a  circuit  having 
a  number  of  pieces  of  apparatus  in  series,  each  of  which 
may  or  may  not  possess  resistance,  inductance,  and 
capacity.  There  can  be  but  one  current  in  that  circuit 
when  a  pressure  is  applied,  and  that  current  must  have 
the  same  phase  throughout  the  circuit.  The  pressure  at 
the  terminals  of  the  various  pieces  of  apparatus,  necessary 
to  maintain  through  them  this  current,  may,  of  course,  be 

of  different  magnitude 
and  in  the  same  or 
different  phases,  being 
dependent  upon  the 
values  of  R,  L,  and 
C.  Therefore  to  de- 
termine the  pressure 
necessary  to  send  a 
certain  alternating  cur- 
rent through  such  a 
series  circuit,  it  is  but 
necessary  to  add  vec- 
torially  the  pressures 
needed  to  send  such  a  current  through  the  separate  parts 
of  the  circuit.  This  is  readily  done  graphically. 

Fig.  35  shows  the  pressures  (according  to  §  22)  neces- 
sary to  send  the  current  /  through  several  pieces  of  ap- 


35- 


PROBLEMS.  51 

paratus,  and  the  combination  of  these  pressures  into  a 
polygon  giving  the  resultant  pressure  E  necessary  to  send 
the  current  /  through  the  several  pieces  in  series.  In 
these  diagrams,  impedance  is  represented  by  the  letter  Z. 
C^  and  C3  are  localized,  not  distributed  capacities. 

For  practical  purposes,  the  quantity  7,  which  is  common 
to  each  side  of  the  triangle,  may  be  omitted  ;  and  merely 
the  impedances  may  be  added  vectorially  in  a  "  polygon  of 
impedances,"  giving  an  equivalent  impedance,  which,  when 
multiplied  by  /,  gives  E. 

Inspection  of  the  figure  shows  that  the  analytical  ex- 
pression for  the  required  E  is 


The  pressure   at   the  terminals  of  any  single  part  of  the 
circuit  is 


It  is  evident  that 

•  EI  +  EI  +  ....>  E, 

and  it  is  found  by  experiment  that  the  sum  of  the  potential 
differences,  as  measured  by  a  voltmeter,  in  the  various 
parts  of  the  circuit,  is  greater  than  the  impressed  pressure. 

27.  A  Numerical  Example  Applying  to  the  Arrangement 
Shown  in  Fig.  35.  —  Suppose  the  pieces  of  apparatus  to 
have  the  following  constants  ; 


ALTERNATING-CURRENT    MACHINES. 


=    85  ohms, 
=    40  ohms, 


=  .25  henry, 
=  .3    henry, 


Ci  =  .000018  farad  (18  mf.) 

Cs  =  .000025  farad, 

RI  =  ioo  ohms.       

With  a  frequency  of  60  cycles  —  whence  (0  =  377 —  it  is 
required  to  find  the  pressure  necessary  to  be  applied  to  the 
circuit  to  send  10  amperes  through  it. 


Fig.  36. 

The  completion  of  the  successive  parallelograms  in 
Fig.  36,  is  equivalent  to  completing  the  impedance  poly- 
gon, and  the  parts  are  so  marked  as  to  require  no  explana- 
tion. The  solution  shows  that  the  equivalent  impedance, 
^=229.5  ohms,  that  the  equivalent  resistance  (=  actual 
resistance  in  series),  R  =  22$  ohms,  that  the  equivalent  re- 
actance is  condensive  and  equals  46.2  ohms,  and  that  </»  = 


PROBLEMS.  53 

11.55°  of  lead.      Hence  the  pressure  required  to  send   10 
amperes  through  the  circuit  is 

E  =  10  x  229.5  =  229S  v°lts. 
To  obtain  the  same  results  analytically 


E  =  10     [85H-4o-fioo]2-f-  [(147.3  ~  94-2)  —  II3-1  4-  io6.2]2, 
E  =  2295  volts. 

The  voltages  at  the  terminals  of  the  various  pieces  of  ap- 
paratus are  : 


l  =  10  V852  -f-  (147.3  ~~  94-2)2  =  iooi  volts, 


£%  =  TO  V4o2  +  1  13.  i2  =1200 

Ez  =  10  Vo2  +  io6.22  =1062 

E±  =  10  Vioo2  4-  o2  =  1000 
El  4-  £2  4-  Ez  +  E± 


which  is  greater  than  .£"  =  2195  volts,  showing  that  the 
numerical  sum  of  the  pressures  is  greater  than  the  im- 
pressed pressure  ;  while  the  vectorial  sum  of  the  separate 
pressures  is  equal  to  the  impressed  pressure. 

28.  Polygon  of  Admittances.  --  If  a  group  of  several 
impedances,  Z^  Z.»  etc.,  be  connected  in  parallel  to  a 
common  source  of  harmonic  E.M.F.  of  E  volts,  their 
equivalent  impedance  is  most  easily  determined  by  con- 
sidering their  admittances  Ylt  Y2,  etc.  The  currents  in 
these  circuits  wouid  be 


The  total  current,  supplied  by  the  source,  would  be  the 
vector  sum  of  these  currents,  due  consideration  being  given 
to  their  phase  relations.  Calling  this  current  /,  the  equation 
I=EY  can  be  written,  where  Fis  the  equivalent  admit- 


54 


ALTERNATING-CURRENT    MACHINES. 


tance  of  the  group.  To  determine  F,  a  geometrical  addition 
of  F1?  F2,  etc.,  must  be  made,  the  angular  relations  being 
the  same  as  the  phase  relations  of  /t,  1^  etc.,  respectively. 
The  value  of  the  equivalent  admittance  may  therefore  be 
represented  by  the  closing  side  of  a  polygon,  whose  other 
sides  are  represented  in  magnitude  by  the  several  admit- 
tances Ylf  F2,  etc.,  and  whose  directions  are  determined 
by  the  phase  angles  of  the  currents  Ilt  /.„  etc.,  flowing 
through  the  admittances  respectively.  The  equivalent  im- 
pedance then  is  equal  to  the  reciprocal  of  F.  The  sum 
of  the  instantaneous  values  of  the  currents  in  the  branch 
circuits  is  equal  to  the  corresponding  instantaneous  values 


Fig.  37- 

in  the  supply  main.  As,  however,  the  maximums  occur 
at  various  times,  the  sum  of  the  effective  currents  in  the 
branches  is  generally  greater  than  the  main  supply  current. 

Fig.  37  is  a  polygon  of  admittances,  showing  the 
method  of  obtaining  the  admittance  F  and  its  phase  angle 
referred  to  a  datum  line,  which  is  equivalent  to  a  number 
of  parallel  admittances,  FT,  F,,  and  F3,  with  angles  fa-fa, 
and  <f>3,  respectively. 

By  taking  its  reciprocal,  the  equivalent  admittance 
can  be  transformed  into  the  equivalent  impedance.  A 
convenient  dimensional  scale  should  be  employed.  The 
impedance  may  be  resolved  into  its  equivalent  reactance 


PROBLEMS. 


and  its  equivalent  resistance.     The  equi- 
valent resistance  is  not  the  resistance  of 

the   parallel    arrangement   as   measured 

by  direct-current   methods. 

As  a  numerical  example,  consider  the 

same  apparatus  as  was  used  in  the  pre-  Fig.  38. 

ceding  example,  §  27,  to  be  arranged  in  parallel,  as  in  Fig. 

38.     All   the  other   conditions  and  values   are  as   stated 

before.  It  is  re- 
quired to  find  the 
current  that  will' 
flow  through  the 
mains  when  ten 
volts  are  impressed 
on  the  circuit.  The 
diagram,  Fig.  39,  is 
self-explanatory. 
The  solution  shows 
Fig-  39-  that  the  equivalent 

admittance    ^=.0224    and   that    <£  =  16.1°.      From  this 

the  equivalent  impedance 

Z  =  —  —  =  44.6  ohms, 

.0224 

the  equivalent  reactance 

[<oZ -    =  Zs'm  d>  =  12.4  ohms, 
0)CJ 

and  the  equivalent  resistance  R  =  Z  cos  4>  =  42.9  ohms. 
The  current  that  will  flow  under  a  pressure  of  10 
volts  is 

I  =  EY=  10  x. 02  24  =  .224  amperes. 


56  ALTERNATING-CURRENT    MACHINES. 

If  a  circuit  have  some  impedances  in  series  and  some 
in  parallel,  or  in  any  series  parallel  combination,  the 
equivalent  impedance  can  always  be  found  by  finding  the 
equivalent  impedances  of  the  several  groups,  and  then 
combining  these  equivalent  impedances  to  get  the  total 
equivalent  impedance  sought. 


ALTERNATORS.  57 


CHAPTER  V. 

ALTERNATORS. 

29.  Single-phase  Alternators As  is  the  case  with 

direct-current  machines,  alternators  have  a  field  and  an 
armature.  The  direct-current  machine's  commutator  is 
replaced,  in  the  single-phaser,  by  a  pair  of  slip-rings ;  an$ 
the  current,  instead  of  being  rectified,  is  lead  out  as 
alternating  current  by  brushes  playing  on  the  rings,  as 
described  in  §  29,  vol.  i.  Revolving  field  and  inductor 
alternators  differ  from  this  arrangement,  as  will  be  shown 
hereafter. 

It  is  necessary  that  all  but  the  very  smallest  alternators 
should  be  multipolar  to  fit  them  to  commercial  require- 
ments. For  alternators  must  have  in  general  a  frequency 
between  25  and  125  cycles  per  second;  the  armature 
must  be  large  enough  to  dissipate  the  heat  generated  at 
full  load  without  its  temperature  rising  high  enough  to 
injure  the  insulation ;  and  finally,  the  peripheral  speed  of 
the  armature  cannot  safely  be  made  to  greatly  exceed  a 
mile  a  minute.  With  these  restrictions  in  mind,  and 
knowing  that  a  point  on  the  armature  must  pass  under 
two  poles  for  each  cycle,  it  becomes  evident  that  alterna- 
tors of  anything  but  the  smallest  capacity  must  be  multi- 
polar. 

In  practice  it  is  quite  as  common  to  have  the  field  of 
an  alternator  revolve  inside  the  armature  as  to  have  the 


58  ALTERNATING-CURRENT   MACHINES. 

armature  revolve.  In  a  few  instances,  notably  at  Niagara, 
the  fields  revolve  outside  the  armature.  The  chief  advan- 
tage of  the  revolving  field  is  that  it  avoids  the  collection 
of  high-tension  currents  through  brushes,  since  the  arma- 
ture may  be  permanently  connected  up,  and  only  low- 
tension  direct  current  need  be  fed  through  the  rings  to 
the  field.  Other  advantages  are  increased  room  for  arma- 
ture insulation,  and,  in  polyphasers,  the  necessity  for  only 
two  instead  of  three  or  more  slip-rings. 

30.  Polyphase  Alternators Single-phase  currents  are 

satisfactory  for  lights,  but  not,  as  yet,  for  power.  As 
polyphase  currents  are  equally  well  adapted  to  both  pur- 
poses, and  since  they  are  generally  more  economical  of 
transmission  than  the  single-phase,  they  are  much  more 
generally  employed.  If  a  motor  be  operated  on  a  single- 
phase  circuit,  the  supply  of  power  to  it  is  pulsating. 
These  pulsations  occur  with  great  rapidity,  there  being  in 
the  case  of  unit  power  factor  two  for  each  cycle.  A 
single-phase  motor  must  be  larger  for  the  same  capacity 
than  a  polyphase  motor. 

Windings  for  any  number  of  circuits  or  phases  may  be 
placed  on  a  single-armature  core,  and  these  may  each  be 
separately  connected  to  an  outside  circuit  through  slip- 
rings,  or  they  may  be  connected  together  in  the  armature 
according  to  some  scheme  whereby  one  slip-ring  will  be 
common  to  two  phases.  These  windings  can  be  placed  so 
that  the  E.M.F.'s  generated  therein  will  have  any  desired 
phase  relations  with  each  other.  It  is  customary  to  place 
them  so  that  the  E.M.Fls  of  a  two-phase  or  four-phase 
system  are  90°  apart,  of  a  three-phase  system  are  120° 
apart,  of  a  six- phase  system  are  60°  apart. 


ALTERNATORS.  $9 

In  the  following  diagrams  the  curled  lines  are  supposed 
to  represent  armature  windings,  which  revolve  in  a  bipolar 
field.  In  some  cases  they  'are  supposed  to  be  wound  on 
cores  so  as  to  form  pole  armatures  and  in  the  other  cases 
to  form  ring  armatures.  The  dots  at  the  terminals  repre- 
sent points  of  transition  between  slip-rings  and  brushes, 
which  are  in  connection  with  line  wires.  It  is  desirable  to 
consider  the  relations  between  the  E.M.F.'s  generated  in 
the  armature  coils  and  the  pressure  between  the  line-wires, 
as  well  as  between  the  currents  in  the  armature  coils  and 
the  currents  in  the  line-wires.  The  assumption  is  made 
that  the  different  phases  are  equally  loaded,  both  as  to 
current  and  as  to  its  phase.  The  system  is  then  said  to 
be  balanced.  It  is  further  assumed  that  the  effective 
E.M.F.  in  each  armature  coil  is  E  volts,  and  the  effective 
current  /  amperes. 

31.   Two-phase  Systems In  the  case  of  two  coils  and 

four  wires,  the    pressure    is   E  volts   between   the  wires 
attached     respectively    to    each    coil. 
There  is  no    connection    between    the 
two  coils  and  their  wires. 

In  case  three  wires  be  employed,  as 
shown  in  Fig.  40,  the  pressure  between 


m  and    n   or   between   /  and    n  is   E    *          Fig>  4°* 
volts,  and  ^2E  volts  between  /  and  m.     I  amperes  flows 
in  /  and  m  and   V2/  in  n. 

32.  Four-phase  or  Quarter-phase  Systems.  — When  con- 
nected, as  in  Fig.  41^,  i.e.,  star  connected,  the  pressure 
between  /and  m  or  n  and  /  is  2E  volts  ;  between  ;/  or/ 
and  /  or  m  it  is  *^2E  volts.  The  current  in  each  line-wire 
is  /  amperes.  If  connected  as  in  Fig  41^,  i.e.,  ring  con-. 


6o 


ALTERNATING-CURRENT   MACHINES. 


nected,  the  pressure  is  E  volts  between  /  and  /z,  n  and  m, 
m  and  /,  or  /  and  /,  and  V2.fi"  volts  between  /  and  m,  or 
n  and  /.  The  current  in  each  line-wire  is  V2/  amperes. 


Star 


Ring 


Fig.  41. 


33.   Three-phase  Systems The  pressure  and  current 

relations  in  three-phase  apparatus  are  often  puzzling  to  the 
student.  Consider  three  similar  coils,  x,  y,  and  2,  on  a 
ring  armature,  each  covering  120°,  as  in  Fig.  42^.  The 
E.M.F.'s  generated  in  these  coils,  when  they  are  rotated 
in  a  bipolar  field,  will  have  the  same  maximum  values,  but 


they  will  differ  in  phase  from  each  other  by  120°.  If  two 
of  the  coils,  x  and  y,  be  connected  as  in  b,  then  the  pres- 
sure between  the  free  terminals  would  be  the  result  of 
adding  the  two  E.M.F.'s  at  120°  with  each  other.  If, 
instead  of  this  connection,  the  one  shown  in  c  be  made, 
known  as  the  star  connection  or  Y  connection,  the  pres- 
sure between  the  free  terminals  would  be  the  result  of 


ALTERNATORS. 


6l 


subtracting  the  E.M.F.   of  coil  y  from  that  of  x  at   120°. 

Subtraction  is  necessary  because  the  connections  of  coil  y 

to  the  circuit   have  been  reversed.     To 

subtract  one  quantity  from  another  it  is 

but  necessary    to   change   its   sign  and 

add.     Therefore   the   pressure  between 

the  free  terminals  is  that  which  results 

from  adding  the  E.M.F .'s  of  x  and  y  at 

300°  (=  120°  +  1 80°)  as  shown  in  Fig. 

43.      It  is    ^I^E  volts.     The  star  connection  is  generally 

represented  as  in  Fig.  44,  where  the  pressure  between  any 

two    line-wires    is    V^fi"  volts,  and   the   current    in    each 
line-wire  is  /  amperes. 

If  the  three  coils  be  connected  as  in 
Fig.  45,  the  result  is  termed  a  delta  (A) 
or  mesh-connection.  The  pressure  be- 
tween any  two  of  the  line-wires  is  E 
volts.  Each  line-wire  is  supplied  with 
current  from  two  coils,  connection  being 

made  at  the  junction  between  the  beginning  of  one  coil 

and  the  ending  of  the  other.     The 

value  of  the  current  in  each  wire 

is  VJ/ amperes.    This  results  from 

subtracting  the  current  in  one  coil 

from   that   in   the  other  at    120°, 

which,  as  before,    is   the  same  as 

adding  the  currents  at  300°. 
The    power  which    is    delivered 

by  a  three-phase   machine   is  not 

altered  by  changing  the  method   of   connection.     In   one 

case  each  phase  is  supplied  with  /  amperes  at   V3  E  volts, 

in  the  other  case  with  \/3  /  amperes  at  E  volts. 


Fig.  44. 


Fig.  45- 


62 


ALTERNATING-CURRENT    MACHINES. 


At  any  instant  the  current  in  one  wire  of  a  three-phase 
system  is  equal  and  opposite  to  the  algebraic  sum  of  the 

currents  in  the  other 
two  wires.  This  is 
clearly  shown  in  Fig. 
46,  where 
found  by 


Fig.  46. 


the    curve 
adding     at 

each  instant  the  ordi- 
nates  of  two  of  the  three-phase  currents  is  similar,  exactly 
equal,  and  opposite  to  the  third  current. 

34.   Electromotive  Force  Generated.  —  In  §  13,  vol.  i.,  it 
was  shown  that  the  pressure  generated  in  an  armature  is 


where 


and 


p  =  number  of  pairs  of  poles; 
<£  =  maxwells  of  flux  per  pole, 
V  =  revolutions  per  minute, 
61  =  number  of  inductors. 


In  an  alternating  current  E  =  k^E^  where  k^  is  the 
form-factor,  i.e.,  the  ratio  of  the  effective  to  the  average 
E.M.F.  Hence  in  an  alternator  yielding  a  sine  wave 
E.M.F.,  v 

£=  2.22p$>S~-  IO-8. 
OO 

v 

Inasmuch  as/  j-  represents  the  frequency,/, 

JS  =   2.22  &SflO~8. 

An  alternator  armature  winding  may  be  either  concen- 
trated or  distributed.  If,  considering  but  a  single  phase, 
there  is  but  one  slot  per  pole,  and  all  the  inductors  that  are 
intended  to  be  under  one  pole  are  laid  in  one  slot,  then 


ALTERNATORS.  63 

the  winding  is  said  to  be  concentrated,  and  if  the  inductors 
are  all  in  series  the  above  formula  for  E  is  applicable.  If 
now  the  inductors  be  not  all  laid  in  one  slot,  but  be  dis- 
tributed in  ;/  more  or  less  closely  adjacent  slots,  the  E.M.F. 

generated  in  the  inductors  of  any  one  slot  will  be  -  of  that 

generated  in  the  first  case,  and  the  pressures  in  the  differ- 
ent slots  will  differ  slightly  in  phase  from  each  other,  since 
they  come  under  the  center  of  a  given  pole  at  different 
times.  The  phase  difference  between  the  E.M.F.  gener- 
ated in  two  conductors  which  are  placed  in  two  successive 
armature  slots,  depends  upon  the  ratio  of  the  peripheral 
distance  between  the  centers  of  the  slots  to  the  peripheral 
distance  between  two  successive  north  poles  considered  as 
360°.  This  phase  difference  angle 

width  slot  +  width  tooth 


—  :  - 

circumference  armature 


"*O 


no.  pairs  poles 

If  the  inductors  of  four  adjacent  slots  be  in  series,  and 
if  the  angle  of  phase  difference  between  the  pressures 
generated  in  the  successive  ones  be  <£,  then  letting  Elt  E^ 
Es,  and  E4  represent  the  respective  pressures,  which  are 


Fig-  47- 


supposed  to  be  harmonic,  the  total  pressure,  E,  generated 
in  them  is  equal  to  the  closing  side  of  the  polygon  as 
shown  in  Fig.  47.  Obviously  E  <  El  +  E.,  -j-  Ez  -\-  Ef 
If  the  winding  had  been  concentrated,  with  all  the  indue- 


64 


ALTERNATING-CURRENT    MACHINES. 


UK) 


.97 


.96 


.94 


,91 


.90 


•action  of  Pole  Dista .. 
Occupiejd  by  SlotsJ 


\ 


1  Slot. 


tors  in  one  slot,  the  total  pressure  generated  would  have 
been  equal  to  the  algebraic  sum. 

The  ratio  of  the  vector  sum  to  the  algebraic  sum  of 
the  pressures  generated  per  pole  and  per  phase  is  called  the 
distribution  constant.  Not  only  may  the  number  of  slots 
under  the  pole  vary,  but  they  may  be  spaced  so  as  to 
occupy  the  whole  surface  of  the  armature  between  succes- 
sive pole  centers  (the  peripheral  distance  between  two 
poles  is  termed  the.  pole  distance),  or  they  may  be  crowded 

together  so  as  to 
occupy  only  one- 
half,  one-fourth,  or 
any  other  fraction 
of  this  space.  Both 
the  number  of  slots 
and  the  fractional 
part  of  the  pole  dis- 
tance which  they 
occupy  affect  the 
value  of  the  distri- 
bution constant.  A 
set  of  curves,  Fig. 
48,  has  been  drawn, 
showing  the  values  of  this  constant  for  various  conditions. 
Curves  are  drawn  for  one  slot  (concentrated  winding),  2, 
3,  4  slots  in  a  group,  and  many  slots  (i.e.,  smooth  core 
with  wires  in  close  contact  on  the  surface).  The  ordinates 
are  the  distribution  constants,  and  the  abscissae  the  frac- 
tional part  of  the  pole  distance  occupied  by  the  slots. 

The  distribution  constant,  k^  must  be  introduced  into 
the  formula  for  the  E.M.F.  giving 


- 

DO 


.3 
Fig.  48. 


2  Slots. 


3  Slots. 

4  Slots. 
Many 
Slots. 


or,  for  sine  waves, 


ALTERNATORS. 


=  2.22 


35.  Armature  Windings.  —  Some  simple  diagrams  of 
the  windings  of  multipolar  alternators  are  given  in  Fig.  49 
et  seq.  The  first  is  a  single-phase  concentrated  winding, 
with  the  winding  which  is  necessary  to  make  it  two-phase 
in  dotted  lines.  If  the  two  windings  be  electrically  con- 
nected where  they  cross  at  point  P  the  machine  becomes 


4-  POLE.  \ 

SINGLE  PHASE.          \ 
CONCENTRATED. 
OF  DOTTED  WINDING  MAKES  IT 
TWO  PHASE 


Fig.  49. 


CONCENTRATED. 

Fig.  50. 


a  star-connected  four-phaser.  Fig.  50  is  a  three-phase, 
A  connected,  concentrated  winding.  Fig.  5  i  is  the  same 
but  y  connected.  The  common  junction  of  the  windings 
would  have  to  be  provided  with  a  slip-ring  if  it  were 
desired  to  operate  a  three-phase,  four-wire  system  with 
the  fourth  wire  connected  to  the  machine.  Fig.  5  2  is  a 
three-phase,  A  connected  winding  distributed  over  two 
slots.  In  all  these  diagrams  the  radial  lines  represent  the 
inductors  ;  other  lines  the  connecting  wires.  The  induc- 
tors of  different  phases  are  drawn  differently  for  clearness. 


66 


ALTERNATING-CURRENT    MACHINES. 


Where  but  one  inductor  is  shown,  in  practice  there  would 
be  a  number  wound  into  a  coil  and  placed  in  the  one  slot. 
For  simplicity  all  the  inductors  of  one  phase  are  shown  in 
series.  In  concentrated  windings,  all  inductors  of  one 


4-  POLE. 
3-  PHASE. 

Y. 
CONCENTRATED. 


Fig.  51. 


phase  carrying  current  in  the  same  direction  could  be 
connected  in  multiple  if  desired  ;  but  with  distributed  wind- 
ings, the  coils  cannot  all  be  placed  in  multiple,  because  the 
small  phase  differences  between  them  would  set  up  local 
currents  and  .give  rise  to  undue  heating. 

To  determine  the  interior  connections  for  a  three-phase 
A  winding,  place  the  inductors  of  a  coil  of  one  phase  under 
the  centers  of  the  poles,  then  a  maximum  pressure  in  a 
given  direction  is  generated,  therein.  Since  the  algebraic 
sum  of  the  pressures  around  the  A  must  be  zero,  the  other 
two  phases  must  be  connected  so  that  their  pressures 
oppose  the  first.  To  determine  the  y  connection,  place 
the  inductors  of  one  phase  under  the  centers  of  the  poles. 
The  E.M.F.  of  this  phase  will  now  be  at  a  maximum,  say, 
away  from  the  common  center.  The  other  two  phases 


ALTERNATORS.  6/ 

must  be  so  connected  as  to  have  E.M.F.'s  toward  the  com- 
mon center  at  this  instant. 

36.   Armature  Reaction The  armature  reaction  of  an 

alternator  consists  of  two  parts,  distortion  and  magnetiza- 
tion or  demagnetization.  These  depend  upon  the  arma- 
ture ampere-turns  and  upon  the  lag  or  lead  of  the  armature 
current.  The  maximum  pres- 
sure is  generated  in  a  coil  when 
its  opposite  inductors  are  re- 
spectively under  the  centers  of 
north  and  south  poles.  This  con- 
dition is  represented  in  Fig.  53.  Fig<  53* 
If  the  armature  current  be  in  phase  with  the  pressure,  /„, 
in  the  coils  coincides  with  Em,  and  poles  on  the  armature 
are  formed  as  shown.  It  is  seen  that  the  M.M.F.'s  both 
of  the  field  and  of  the  armature  conspire  to  concentrate 
the  flux  in  the  trailing  pole  tips.  So  with  /  in  phase  with 
E  the  armature  M.M.F.  chiefly  effects  a  distortion  of  the 
lines,  entailing  a  greater  flux  density,  hence  a  lower  per- 
meability, and  also  a  greater  length  of  air-gap  path.  This 
slightly  decreases  the  flux,  and  affects  the  regulation  of 
the  alternator. 

If,  now,  the  current  be  lagging,  the  armature  will  have 
reached  a  position  in  advance,  at  the  instant  of  maximum 
current.  Therefore,  like  poles  of  the  field  and  of  the 
armature  will  be  more  directly  opposite  to  each  other. 
The  distorting  influence  will  be  present  in  a  degree ;  and 
there  will  be  considerable  demagnetization  of  the  field,  due 
to  the  opposing  M.M.F.'s  of  the  armature  and  field  ampere- 
turns.  If  the  current  be  leading,  then,  at  the  instant  of 
maximum  current,  a  south,  armature  pole  will  be  more 


68  ALTERNATING-CURRENT    MACHINES. 

nearly  opposite  to  a  north  field  pole,  and  their  M.M.F.'s 
will  be  cumulative.  The  field  will  be  strengthened  if  the 
magnetizing  reaction  exceeds  in  effect  the  skewing  reac- 
tion. Alternators  have  a  much  better  regulation  on  non- 
inductive  loads  than  on  inductive  loads. 

37.  Armature  Inductance The  impedance  of  an  alter- 
nator armature  is  made  up  of  its  ohmic  resistance,  R, 
combined  at  right  angles  with  its  reactance,  2  -nfL.  In 
practice  the  inductance,  Z,  is  likely  to  be  so  great  that  R 
becomes  negligible,  and  the  impedance  equals  the  reac- 
tance. The  armature  reactance  may  or  may  not  be  an 
appreciable  part  of  the  impedance  offered  by  the  completed 
circuit.  If  it  is  appreciable,  then  the  current  in  the  circuit 
will  lag  even  with  a  non-inductive  load.  In  any  case  there 
will  be  loss  of  voltage  due  to  armature  impedance  which 
(when  R  is  negligible)  is  equal  to  2  -of LI.  This  is  at  right 
angles  to  the  current,  and  must  be  properly  combined  with 
/  times  the  equivalent  impedance  of  the  external  cir- 
cuit to  determine  the  pressure  actually  generated  in  the 
machine.  In  special  cases  the  armature  reactance  is  the 
predominant  feature  of  the  circuit ;  for  instance,  alternators 
for  series  arc  lighting  are  made  with  so  great  a  reactance 
that  the  impedance  of  the  external  circuit  within  the 
limits  of  operation  is  negligible  in  comparison.  The  altera- 
tion in  the  value  of  this  impedance  does  not,  then,  appre- 
ciably alter  the  total  impedance  of  the  circuit,  and  the 
alternator  therefore  operates  as  a  constant-current  gen- 
erator. Many  commercial  alternators  have  sufficient  arma- 
ture reactance  to  prevent  their  injuring  themselves  on 
dead  short  circuit  for  a  limited  time.  It  is  necessary 
that  armatures  should  have  some  considerable  inductance 


ALTERNATORS.  69 

when    alternators    are    to    be    operated    satisfactorily   in 
parallel. 

38.  Synchronous  Reactance When  an  alternator  is  op- 
erating on  a  load,  the  pressure,  which  would  be  generated 
on  open  circuit  at  the  same  speed  and  excitation,  is  made 
up  of  the  following  parts,  and  might  be  found  by  adding 
them  together  in  their  proper  phase  relations : 

(a)  terminal  voltage,  E, 

(b)  ohmic  drop  in  armature,  IR,  in  phase  with  the  cur- 
rent, 

(c)  armature  inductance  drop,  90°  with  the  current, 

(d)  deficit  of  actually  generated  volts  due  to  increase  of 
magnetic  reluctance  accompanying  distortion, 

(e)  deficit  or  increment  of  actually  generated  volts  due 
to  the  demagnetization  of  a  lagging  current   or  the  mag- 
netization of  a  leading  current. 

All  the  parts,  except  the  first  mentioned,  can  be  grouped 
together,  and  be  dealt  with  collectively  by  the  use  of  a 
quantity  called  the  synchronous  impedance.  It  is  that  im- 
pedance, which,  if  connected  in  series  with  the  outside  cir- 
cuit and  an  impressed  voltage  of  the  same  value  as  the 
open-circuit  voltage  at  the  given  speed  and  excitation,  would 
permit  a  current  of  the  same  value  to  flow  as  does  flow. 
This  quantity  for  any  load  can  be  determined  experimen- 
tally with  ease.  The  synchronous  impedance  has  two  fac- 
tors, namely,  the  armature  resistance  and  a  quantity  termed 
the  synchronous  reactance.  The  two,  when  combined  at 
right  angles,  give  the  synchronous  impedance. 

Since  the  synchronous  impedance  takes  account  of  all 
the  diverse  causes  of  voltage  drop  above  enumerated,  it  is 
clear  that  it  has  not  a  physical  existence,  but  is  merely  a 


70  ALTERNATING-CURRENT   MACHINES. 

fiction.  It  is  of  great  use  in  determining  the  performance 
of  a  machine.  Its  value  is  the  same  for  all  excitations  of 
the  field,  but  is  somewhat  different  for  various  loads. 
These  two  facts  afford  a  very  convenient  means  of  deter- 
mining its  value.  Run  the  alternator  at  its  proper  speed. 
Short-circuit  the  armature  through  an  ammeter.  Excite 
the  field  until  the  ammeter  indicates  the  desired  load. 
Then  open  the  load  circuit  and  read  the  terminal  voltage. 
The  quotient  of  the  volts  by  the  amperes  is  the  synchron- 
ous impedance.  It  may  happen  that  the  resistance  of  the 
armature  is  negligibly  small,  in  which  case  the  synchron- 
ous reactance  equals  the  synchronous  impedance. 

39.  Saturation  Coefficient —  A  no-load  saturation  curve 
of  an  alternator  may  be  obtained  by  measuring  the  termi- 
nal voltage  corresponding  to  various  strengths  of  field  cur- 
rent, when  the  machine  is  running  at  its  proper  speed  and 

without  load.  Laying  off 
E.M.F.'s,  E  as  ordinates 
and  exciting  currents,  If) 
as  abscissae,  a  curve  is 
found  as  in  Fig.  54. 

The  ratio   —£-   is   called 
dE 

E 

Excitation  the  no-load  saturation  co- 


FUE.  54.  efficient  of  the  machine. 

Another  curve,   called  the  load -saturation  curve  can   be 

obtained  by  using  a  variable  non-inductive  resistance  for 

maintaining   the   constant   full   load.     The   terminal  volts 

corresponding  to  various   field  excitations  are  read  on  a 


ALTERNATORS.  71 

voltmeter.  This  curve  will  approximately  parallel  the  no- 
load  saturation  curve.  It  will  have  a  zero  voltage  value 
for  that  excitation  which  causes  sufficient  voltage  to  send 
the  full-load  current  through  the  synchronous  impedance 
of  the  armature.  A  full-load  saturation  coefficient  curve 
might  be  drawn  from  the  full-load  saturation  curve.  It 
will  nearly  coincide  with  the  other  coefficient  curve. 

These  saturation  curves  have  forms  similar  to  magneti- 
zation curves  for  iron.  The  knee,  however,  is  less  abrupt 
than  is  general  in  an  iron  curve,  because  of  the  unvarying 
permeability  of  air,  and  because  the  different  magnetic 
parts  of  the  generator  do  not  reach  saturation  at  the  same 
time.  If  the  alternator  is  normally  excited  to  above  the 
knee  of  the  saturation  curve,  it  will  require  a  considerable 
increase  of  field  current  to  maintain  the  terminal  voltage 
when  the  load  is  thrown  on,  while  if  normally  excited 
below  the  knee,  a  slight  increase  of  excitation  will  suffice. 
The  regulation  is,  however,  better  when  the  magnetization 
is  above  the  knee ;  that  is,  with  unaltered  field  strength, 
the  voltage  rise  upon  throwing  off  the  load  is  less  than  if 
the  excitation  were  below  the  knee. 

40.  Leakage  Coefficient As  in  direct-current  machines, 

the  leakage  coefficient  of  an  alternator  may  be  defined  as 
the  number  of  maxwells  set  up  by  the  field  divided  by  the 
number  of  maxwells  passing  through  the  armature.  It  is 
always  greater  than  unity.  Its  value  depends  upon  the 
design  of  the  machine,  upon  the  permeability  of  the  various 
parts  making  up  the  magnetic  circuit,  upon  the  load  on 
the  machine,  and  upon  the  degree  of  saturation  in  the 
fields.  In  modern  commercial  machines  of  size  its  values 
lie  between  i.i  and  1.5. 


72  ALTERNATING-CURRENT   MACHINES. 

41.  Efficiency The  following  is  abstracted  from  the 

Report  of  the  Committee  on  Standardization  of  the  Ameri- 
can Institute  of  Electrical  Engineers.  Only  those  por- 
tions are  given  which  bear  upon  the  efficiency  of  alternators. 
They  will,  however,  apply  equally  well  to  synchronous 
motors. 

The  "  efficiency"  of  an  apparatus  is  the  ratio  of  its  net 
power  output  to  its  gross  power  input. 

Electric  power  should  be  measured  at  the  terminals  of 
the  apparatus. 

In  determining  the  efficiency  of  alternating-current 
apparatus,  the  electric  power  should  be  measured  when 
the  current  is  in  phase  with  the  E.M.F.  unless  otherwise 
specified,  except  when  a  definite  phase  difference  is  in- 
herent in  the  apparatus,  as  in  induction  motors,  etc. 

Where  a  machine  has  auxiliary  apparatus,  such  as  an 
exciter,  the  power  lost  in  the  auxiliary  apparatus  should 
not  be  charged  to  the  machine,  but  to  the  plant  consisting 
of  the  machine  and  auxiliary  apparatus  taken  together. 
The  plant  efficiency  in  such  cases  should  be  distinguished 
from  the  machine  efficiency. 

The  efficiency  may  be  determined  by  measuring  all  the 
losses  individually,  and  adding  their  sum  to  the  output  to 
derive  the  input,  or  subtracting  their  sum  from  the  input 
to  derive  the  output.  All  losses  should  be  measured  at, 
or  reduced  to,  the  temperature  assumed  in  continuous 
operation,  or  in  operation  under  conditions  specified. 

In  synchronous  machines  the  output  or  input  should  be 
measured  with  the  current  in  phase  with  the  terminal 
E.M.F.  except  when  otherwise  expressly  specified. 

Owing  to  the  uncertainty  necessarily  involved  in  the 
approximation  of  load  losses,  it  is  preferable,  whenever 


ALTERNATORS.  73 

possible,  to  determine  the  efficiency  of  synchronous  ma- 
chines by  input  and  output  tests. 

The  losses  in  synchronous  machines  are  : 

a.    Bearing  friction  and  windage. 

/;.  Molecular  magnetic  friction  and  eddy  currents  in 
iron,  copper,  and  other  metallic  parts.  These  losses  should 
be  determined  at  open  circuit  of  the  machine  at  the  rated 
speed  and  at  the  rated  voltage,  +  IR  in  a  synchronous 
generator,  —  IR  in  a  synchronous  motor,  where  /  =  cur- 
rent in  armature,  R  =  armature  resistance.  It  is  undesir- 
able to  compute  these  losses  from  observations  made  at 
other  speeds  or  voltages. 

These  losses  may  be  determined  by  either  driving  the 
machine  by  a  motor,  or  by  running  it  as  a  synchronous 
motor,  and  adjusting  its  fields  so  as  to  get  minimum  cur- 
rent input,  and  measuring  the  input  by  wattmeter.  The 
former  is  the  preferable  method,  and  in  polyphase  ma- 
chines the  latter  method  is  liable  to  give  erroneous  results 
in  consequence  of  unequal  distribution  of  currents  in  the 
different  circuits  caused  by  inequalities  of  the  impedance 
of  connecting  leads,  etc. 

c.  Armature-resistance    loss,  which   may  be    expressed 
by/  I^R  ;  where  R  =  resistance  of  one  armature  circuit 
or  branch,  /  =  the  current    in    such  armature    circuit   or 
branch,    and  p  =  the    number    of    armature     circuits    or 
branches. 

d.  Load  losses.     While    these    losses    cannot   well    be 
determined   individually,   they  may  be   considerable,   and, 
therefore,  their  joint  influence  should    be  determined  by 
observation.     This  can  be  done  by  operating  the  machine 
on  short  circuit  and  at  full-load  current,  that  is,  by  deter- 
mining what  may  be  called  the  "short-circuit  core  loss." 


74  ALTERNATING-CURRENT    MACHINES. 

With  the  low  field  intensity  and  great  lag  of  current 
existing  in  this  case,  the  load  losses  are  usually  greatly 
exaggerated. 

One-third  of  the  short-circuit  core  loss  may,  as  an 
approximation,  and  in  the  absence  of  more  accurate  infor- 
mation, be  assumed  as  the  load  loss. 

e.  Collector-ring  friction  and  contact  resistance.     These 
are  generally  negligible,  except  in  machines  of  extremely 
low  voltage. 

f.  Field    excitation.      In    separately    excited    machines, 
the  I^R  of  the  field  coils  proper  should  be  used.     In  self- 
exciting  machines,  however,  the  loss  in  the  field  rheostat 
should  be  included. 

42.  Regulation  for  Constant  Potential.  —  Alternators 
feeding  light  circuits  must  be  closely  regulated  to  give 
satisfactory  service.  The  pressure  can  be  maintained 
constant  in  a  circuit  by  a  series  boosting  transformer,  but 
it  is  generally  considered  better  to  regulate  the  dynamo  by 
suitable  alteration  of  the  field  strength. 

The  simplest  method  of  regulating  the  potential  is  to 
have  a  hand-operated  rheostat  in  the  field  circuit  of  the 
alternator,  when  the  latter  is  to  be  excited  from  a  com- 
mon source  of  direct  current,  or  in  the  field  circuit  of  the 
exciter,  if  the  alternator  is  provided  with  one.  The 
latter  method  is  generally  employed  in  large  machines, 
since  the  exciter  field  current  is  small,  while  the  alternator 
field  current  may  be  of  considerable  magnitude,  and  would 
give  a  large  I2R  loss  if  passed  through  a  rheostat. 

A  second  method  of  regulation  employs  a  composite 
winding,  analogous  to  the  compound  windings  of  direct- 
current  generators.  This  consists  of  a  set  of  coils ;  one 


ALTERNATORS. 


75 


on  each  pole.  These  are  connected  in  series,  and  carry 
a  portion  of  the  armature  current  which  has  been  rectified. 
The  rectifier  consists  of  a  commutator,  having  as  many 
segments  as  there  are  field  poles.  The  alternate  segments 
are  connected  together,  forming  two  groups.  The  groups 
are  connected  respectively  with  the  two  ends  of  a  resis- 
tance forming  part  of  the  armature  circuit.  Brushes, 


/<S  fl-60-900  Form  /q 
^S  0-90-900  Torn-,  f\ 
>^S  CH20-300  Form  rt 


Commutator-Collector . 


manner  of  placing  spools 
Tne  observer  f&  supposed  to  be  looking  at  face 


winding  snoul. 
toward  the  ol 


ase^on  spool  f  langes.tr 

i-i-wsucceeding  spool.  „, 

Fig.  55- 


being  so  placed 
' 


bearing  upon  the  commutator,  connect  with  the  terminals 
of  the  composite  winding.  The  magnetomotive  force  of 
the  composite  winding  is  used  for  regulation  only,  the 
main  excitation  being  supplied  by  an  ordinary  separately 
excited  field  winding.  The  rectified  current  in  the  com- 
posite coils  is  a  pulsating  unidirectional  current,  that 
increases  the  magnetizing  force  in  the  fields  as  the  cur- 
rent in  the  armature  increases.  The  rate  of  increase  is 


76  ALTERNATING-CURRENT    MACHINES. 

determined  by  the  resistance  of  a  shunt  placed  across  the 
brushes.  By  increasing  the  resistance  of  this  shunt,  the 
amount  of  compounding  can  be  increased.  With  such 
an  arrangement  an  alternator  can  be  over-compounded  to 
compensate  for  any  percentage  of  potential  drop  in  the 
distributing  lines.  The  method  here  outlined  is  used  by 
the  General  Electric  Company  in  their  single-phase 
stationary  field  alternators.  The  connections  are  shown 
in  Fig.  55, 

A  third  method  of  regulation  is  employed  by  the  West- 
inghouse  Company  on  their  revolving  armature  alter- 
nators, one  of  which,  a  75  K.W.,  60^,  single-phase  machine, 
is  shown  in  Fig.  56.  A  composite  winding  is  employed, 
and  the  compensating  coils  are  excited  by  current  from 
a  series  transformer  placed  on  the  spokes  of  the  armature 
spider.  The  primary  of  this  transformer  consists  of  but 
a  few  turns,  and  the  whole  armature  current  is  conducted 
through  it  before  reaching  the  collector  rings.  The  sec- 
ondary of  this  transformer  is  suitably  connected  to  a 
simple  commutator  on  the  extreme  end  of  the  shaft. 
Upon  this  rest  the  brushes  which  are  attached  to  the  ends 
of  the  compensating  coil.  This  commutator  is  subjected 
to  only  moderate  currents  and  low  voltages.  The  current 
in  the  secondary  of  the  transformer,  and  hence  that  in 
the  compensating  coil,  is  proportional  to  the  main  armature 
current.  The  machine  is  wound  for  the  maximum  desir- 
able over-compounding,  and  any  less  compensation  can  be 
secured  by  slightly  shifting  the  commutator  brushes. 
For  there  are  only  as  many  segments  as  poles  ;  and  if  the 
brushes  span  the  insulation  just  when  the  wave  of  current 
in  the  transformer  secondary  is  passing  through  zero, 
then  the  pulsating  direct  current  in  the  compounding  coil 


ALTERNATORS. 


77 


is  equal  to  the  effective  value  of  the  alternating  current ; 
but  if  the  brushes  are  at  some  other  position,' the  current 
will  flow  in  the  field  coil  in  one  direction  for  a  portion  of 
the  half  cycle,  and  in  the  other  direction  for  the  remaining 


Fig.  56. 


portion.  A  differential  action,  therefore,  ensues,  and  the 
effective  value  of  the  compensating  current  is  less  than  it 
was  before. 

In    order    to  produce  a  constant    potential    on   circuits 
having  a  variable  inductance  as  well  as  a  variable  resist- 


78  ALTERNATING-CURRENT    MACHINES. 

ance,  the  General  Electric  Co.  has  designed  its  compensated 
revolving  field  generators,  which  are  constructed  for  two- 
or  three-phase  circuits.  The  machine,  Fig.  57,  is  of  the 


Fig.  57- 

revolving  field  type,  the  field  being  wound  with  but  one 
simple  set  of  coils.  On  the  same  shaft  as  the  field,  and 
close  beside  it,  is  the  armature  of  the  exciter,  as  shown 
in  Fig.  58.  The  outer  casting  contains  the  alternator 
armature  windings,  and  close  beside  them  the  field  of  the 
exciter.  This  latter  has  as  many  poles  as  has  the  field  of 
the  alternator.  Alternator  and  exciter,  therefore,  operate 
in  a  synchronous  relation.  The  armature  of  the  exciter  is 
fitted  with  a  regular  commutator,  which  delivers  direct 
current  both  to  the  exciter  field  and,  through  two  slip- 


ALTERNATORS. 


79 


rings,  to  the  alternator  field.  On  the  end  of  the  shaft, 
outside  of  the  bearings,  is  a  set  of  slip-rings,  four  for  a 
quart  er-phaser,  three  for  a  three-phaser,  through  which 
the  exciter  armature  receives  alternating  current  from  one 
or  several  series  transformers  inserted  in  the  mains  which 
lead  from  the  alternator.  This  alternating  current  is 
passed  through  the  exciter  armature  in  such  a  manner  as 
to  cause  an  armature  reaction,  as  described  in  §  36,  that 
increases  the  magnetic  flux.  This  raises  the  exciter  vol- 
tage and  hence  increases  the  main  field  current.  The 


Fig.  58. 

reactive  magnetization  produced  in  the  exciter  field  is 
proportional  to  the  magnitude  and  phase  of  the  alternating 
current  in  the  exciter  armature.  The  reactive  demag- 
netization of  the  alternator  field  is  proportional  to  the 
magnitude  and  phase  of  the  current  in  the  alternator 
armature.  And  these  currents  have  the  fixed  relations 
of  current  strength  and  phase,  which  are  determined  by  the 
series  transformers.  Hence  the  exciter  voltage  varies  so 
as  to  compensate  for  any  drop  "in  the  terminal  voltage. 
Neither  the  commutator  nor  any  of  the  slip-rings  carry 
pressures  of  over  75  volts.  The  amount  of  over-corn- 


8o 


ALTERNATING-CURRENT   MACHINES. 


pounding  is  determined  by  the  ratio  in  the  series  trans- 
formers. The  normal  voltage  of  the  alternator  may  be 
regulated  by  a  small  rheostat  in  the  field  circuit  of  the 
exciter. 

43.    Inductor   Alternators Generators  in  which  both 

armature  and  field  coils  are  stationary  are  called  inductor 
alternators.  Fig.  59  shows  the  principle  of  operation  of 


/ARMATURE  COILS 


Fig.  59- 

these  machines.  A  moving  member,  carrying  no  wire, 
has  pairs  of  soft  iron  projections,  which  are  called  induc- 
tors. These  projections  are  magnetized  by  the  current 
flowing  in  the  annular  field  coil  as  shown  in  figure.  The 
surrounding  frame  has  internal  projections  corresponding 
to  the  inductors  in  number  and  size.  These  latter  projec- 
tions constitute  the  cores  of  armature  coils.  When  the 
faces  of  the  inductors  are  directly  opposite  to  the  faces  of 
the  armature  poles,  the  magnetic  reluctance  is  a  minimum, 
and  the  flux  through  the  armature  coil  accordingly  a  maxi- 
mum. For  the  opposite  reason,  when  the  inductors  are 
in  an  intermediate  position  the  flux  linked  with  the  arma- 
ture coils  is  a  minimum.  As  the  inductors  revolve,  the 
linked  flux  changes  from  a  maximum  to  a  minimum,  but  it 
does  not  change  in  sign. 

Absence  of  moving  wire  and  the  consequent  liability  to 


ALTERNATORS. 


81 


chafing   of   insulation,  absence  of    collecting  devices  and 
their  attendant  brush  friction,  and  increased  facilities  for 


Fig.  60. 

insulation  are  claimed  as  advantages  for  this  type  of  ma- 
chine.    By  suitably  disposing  of  the  coils,  inductor  alter- 
__^^  nators  may  be  wound 

for    single-    or    poly- 
y     phase  currents. 

The  Stanley  Elec- 
tric Manufacturing 
Company  manufac- 
ture two- phase  induc- 
tor alternators.  A 
view  of  one  of  their 
machines  is  given  in 
Fig.  61.  Fig.  60,  with  the 

frame  separated  for  inspection  of  the  windings.     In  this 
picture  the  field  coil  is  hanging  loosely  between  the  pairs 


82 


ALTERNATING-CURRENT    MACHINES. 


of-  inductors.  The  theoretical  operation  of  this  machine  is 
essentially  that  described  above.  All  iron  parts,  both 
stationary  and  revolving,  that  are  subjected  to  pulsations 
of  magnetic  flux,  are  made  up  of  laminated  iron.  The 


Fig.  6aa. 


Fig.  63b. 


84  ALTERNATING-CURRENT   MACHINES. 

large  field  coil  is  wound  on  a  copper   spool.     Ordinarily 
when  the  field  circuit  of  a  large  generator  is  broken,  the 


Fig.  64. 


E.M.F.  of  self-induction  may  rise  to  so  high  a  value  as 
to  pierce  the  insulation.  With  this  construction  the  cop- 
per spool  acts  as  a  short  circuit  around  the  decaying  flux, 


ALTERNATORS.  85 

and  prevents  high  E.M.F!$>  of  self-induction.  Figs.  61 
and  62  show  the  details  of  construction  of  a  Stanley 
machine  of  a  larger  size  than  the  one  previously  shown. 
Inductor  alternators  are  also  manufactured  by  Westing- 
house  and  Warren  companies.  The  construction  of  the 
machine  made  by  the  latter  company  is  shown  in  Fig.  63^, 
There  is  but  a  single  field  coil,  which  fits  into  the  recess  in 
the  back  of  the  frame  as  shown.  The  armature  coils  sur- 
round the  pole  projections,  and  the  flux  through  them  is 
altered  by  the  change  of  reluctance  caused  by  the  ro- 
tating inductor  which  carries  no  wire.  The  exciter  is 
carried  upon  a  platform  which  (Fig.  63 £)  forms  part  of 
the  main  frame  and  is  driven  by  a  belt  from  a  pulley  on 
the  armature  shaft. 

44.  Revolving  Field  Alternators In  this  type  of  al- 
ternator, the  armature  windings  are  placed  on  the  inside 
of  the  surrounding  frame,  and  the  field  poles  project  radi- 


Fig.  65. 


ally  from  the  rotating  member.  As  was  stated  before,  this 
type  of  construction  is  to  be  recommended  in  the  case  of 
large  machines  which  are  required  to  give  either  high 


86 


ALTERNATING-CURRENT   MACHINES. 


voltages    or    large    currents.      With    the    same   peripheral 
velocity,  there  is  more  space  for  the  armature  coils ;  the 


SCALE  K  INtJH  EQUJ.U6  ONE  FOOT^ 

Fig.  66. 


coils  can  be  better  ventilated,  air  being  forced  through 
ducts  by  the  rotating  field  ;  stationary  coils  can  be  more 
perfectly  insulated  than  moving  ones  ;  and  the  only  cur- 


ALTERNATORS. 


rents  to  be  collected  by  brushes  and  collector  rings  are 
those  necessary  to  excite  the  fields. 

Fig.  64  shows  a  General  Electric  750  K.  w.  revolving 
field  generator.     The  two  collector  rings  for  the  field  cur- 


Field  Current 


.4        .5       .6        J        .8       .9 
Output-Proportion  of  full  load 
Fig.  67. 


1.0      1.1      L2      U3 


rent  are  shown,  and  in  Fig.  65  the  edgewise  method  of 
winding  the  field  colls  is  shown.  The  collector  rings  are 
of  cast  iron  and  the  brushes  are  of  carbon.  Fig.  66  shows 
the  details  of  construction  of  a  5,000  K.  w.  three-phase 


88  ALTERNATING-CURRENT    MACHINES. 


OF 


ALTERNATORS. 


6,6oo-volt  machine  of  this  type  as  constructed  for  the 
Metropolitan  Street  Railway  Co.  of  New  York.  This 
machine  has  40  poles,  runs  at  75  R.  p.  M.  at  a  peripheral 


Fig.  69. 

velocity  of  3,900  feet  per  minute.  This  gives  a  frequency 
of  25.  The  air  gap  varies  from  five-sixteenths  at  the 
pole  center  to  eleven-sixteenths  at  the  tips.  The  short- 
circuit  current  at  full-load  excitation  is  less  than  800  am- 
peres per  leg.  The  rated  full-load  current  is  slightly  over 
300  amperes.  The  efficiency  and  no-load  saturation  curve 
is  shown  in  Fig.  67. 

The  Bullock  Electric  Mfg.  Co.  also  make  generators  of 
this  type.  The  method  of  placing  armature  coils  is  shown 
in  Fig.  68.  These  coils,  as  shown,  are  wire  wound,  taped, 
insulated,  and  held  in  slots  by  maple-wood  wedges.  The 
field  poles  are  fastened  directly  to  a  spider  having  a  heavy 
rim.  The  pole  pieces  are  of  T-shaped  steel  punchings, 


90  ALTERNATING-CURRENT   MACHINES. 


Fig. -70. 


held  together  by  rivets  and  malleable  iron  end  pieces. 
These  are  fastened  to  the  rim  of  the  spider  by  bolts  in  the 
case  of  slow-speed  machines,  or  are  dovetailed  to  fit  slots 
in  the  rim  in  the  case  of  high-speed  machines.  This  latter 


ALTERNATORS.  91 

method  of  fastening  is  shown  in  Fig.  69,  which  represents 
the  field  and  shaft  of  a  small-sized  .high-speed  machine. 

The  Westinghouse  rotating  field  consists  of  a  steel  rim 
mounted  upon  a  cast-iron  spider.  Into  dovetailed  slots  in 
the  rim  are  fitted  laminated  plates  with  staggered  joints. 
These  plates  are  bolted  together.  The  laminations  are 
supplied  at  intervals  with  ventilating  ducts.  The  coils  are 
kept  in  place  by  retaining  wedges  of  non-magnetic  material. 
A  portion  of  a  field  is  shown  in  Fig.  70. 


92  ALTERNATING-CURRENT  MACHINES. 


CHAPTER  VI. 

THE  TRANSFORMER. 

45.  Definitions.  —  The  alternating-current  transformer 
consists  of  one  magnetic  circuit  interlinked  with  two  elec- 
tric circuits,  of  which  one,  the  primary,  receives  electrical 
energy,  and  the  other,  the  secondary,  delivers  electrical 
energy.  If  the  electric  circuits  surround  the  magnetic 
circuit,  as  in  Fig.  71,  the  transformer  is  said  to  be  of  the 

core  type.  If  the  re- 
verse is  true,  as  in 
Fig.  72,  the  trans- 
former is  of  the  shell 
type.  The  practical 
utility  of  the  trans- 
former lies  in  the  fact 
that,  when  suitably  de- 
signed, its  primary  can 
take  electric  energy  at 
one  potential,  and  its 
secondary  deliver  the 

Fig.  71. 

same  energy  at   some 

0ther  potential ;  the  ratio  of  the  current  in  the  primary  to 
that  in  the  secondary  being  approximately  inversely  as  the 
ratio  of  the  pressure  on  the  primary  to  that  on  the  secon- 
dary. 

The  ratio  of  transformation  of  a  transformer  is  repre- 


THE   TRANSFORMER.  93 

sented  by  r,  and  is  the  ratio  of  the  number  of  turns  in  the 
secondary  coils  to  the  number  of  turns  in  the  primary  coil. 
This  would  also  be  the  ratio  of  the  secondary  voltage  to 


Fig.  72. 

the  primary  voltage,  if  there  were  no  losses  in  the  trans- 
former. A  transformer  in  which  this  ratio  is  greater  than 
unity  is  called  a.  "  step-up  "  transformer,  since  it  delivers 
electrical  energy  at  a  higher  pressure  than 'that  at  which 
it  is  received.  When  the  ratio  is  less  than  unity  it  is 
called  a  "  step-down  "  transformer.  Step-up  transformers 
find  their  chief  use  in  generating  plants,  where  because  of 
the  practical  limitations  of  alternators,  the  alternating  cur- 
rent generated  is  not  of  as  high  a  potential  as  is  demanded 
for  economical  transmission.  Step-down  transformers  find 
their  greatest  use  at  or  near  the  points  of  consumption  of 
energy,  where  the  pressure  is  reduced  to  a  degree  suitable 
for  the  service  it  must  perform.  The  conventional  repre- 
sentation of  a  transformer  is  given  in  Fig.  73.  In  general, 
little  or  no  effort  is  made  to  indicate  the  ratio  of  trans- 
formation by  the  relative  number  of  angles  or  loops  shown, 


94  ALTERNATING-CURRENT    MACHINES. 

though    the    low-tension    side  is   sometimes  distinguished 
from  the  high-tension  side  by  this  means. 

When  using  the  same  or  part  of  the  same  electric  cir- 
cuit for  both  primary  and  secondary,  the  device  is  called 
an  auto-transformer.  These  are  sometimes  used  in  the 


1 


Fig.  73-  Fig.  74- 

starting  devices  for  induction  motors,  and  sometimes 
connected  in  series  in  an  alternating-current  circuit,  and 
arranged  to  vary  the  E.M.F.  in  that  circuit.  Fig.  74  is 
the  conventional  representation  of  an  auto-transformer. 

46.  Core  Flux  —  (a)  Open-circuited  secondary.  When 
the  secondary  coil  of  a  transformer  is  open-circuited  it  is 
perfectly  idle,  having  no  influence  on  the  rest  of  the  ap- 
paratus, and  the  primary  becomes  then  merely  a  choke 
coil.  A  transformer  is  so  designed  that  its  reactance  is 
very  high,  and  its  resistance  comparatively  low.  This 
makes  a  large  impedance,  which  is  almost  wholly  reactive ; 
hence  the  current  that  will  flow  in  the  primary  when  the 
secondary  is  open-circuited  is  very  small,  and  lags  practi- 
cally 90°  behind  the  E.M.F.  which  sends  rt.  This 
current  is  called  the  exciting  current,  or  sometimes  less 
properly  the  magnetizing  current  or  leakage  current.  A 
flux  is  set  up  in  the  iron  of  a  transformer,  which  is  sinu- 
soidal and  is  in  phase  with  the  exciting  current.  This  flux 
induces  a  practically  sinusoidal  E.M.F.  in  the  primary 
coil,  90°  behind  it  in  phase ;  because  the  induced  E.M.F. 
is  greatest  when  the  time  rate  of  flux  change  is  greatest, 
and  the  flux  changes  fastest  as  it  is  passing  through  the 


THE   TRANSFORMER.  95 

zero  value.  This  induced  E.M.F.  is  90°  behind  the  flux, 
which  in  turn  is  90°  behind  the  impressed  pressure ;  there- 
fore the  induced  E.M.F.  is  180°  behind  the  impressed 
E.M.F.  or  is  a  counter  E.M.F.  The  counter  pressure  is 
less  than  the  impressed  pressure  only  by  the  small  amount 
necessary  to  cause  the  small  exciting  current  to  flow. 
Neglecting  the  primary  resistance,  Rp,  and  the  reluctance, 
(R,  of  the  core,  the  counter  pressure  would  be  equal  to  the 
impressed  pressure  ;  and  in  commercial  transformers  this 
is  true  to  within  a  small  percentage.  Considering  that 
the  flux  varies  sinusoidally,  and  that  its  maximum  value 
is  3>m ;  then  the  flux  at  any  time,  /,  is  <£OT  cos  w/,  and  the 
counter  E.M.F.,  which  is  equal  and  opposed  to  the  im- 
pressed primary  pressure  Et>,  may  be  written  (§  13,  vol.  i.) 

„  f       nv  </(<£TOcos  w/) 

E*=7J W~-' 

and  since  4>w  and  w  are  constant 

Ep  =  i  o~~8  «po>4>w  sin  w/, 
from  which 

lO8^,,,,  1C 


and 


This  equation  is  used  in  designing  transformers  and 
choke  coils.  The  values  of  <&m  for  60  cycle  transformers 
of  different  capacities,  as  determined  by  experiment  and 
use,  are  shown  in  the  curve,  Fig.  75.  It  is  usual  in  such 
designs  to  also  assume  a  maximum  flux  density,  (Bm. 
While  the  value  assumed  differs  much  with  different  man- 
ufacturers, it  is  safe  to  say  that  for  25  cycles  (Bw  varies 
between  9  and  14  kilogausses  ;  for  60  cycles  between  6 
and  9  kilogausses;  and  for  125  cycles  between  5  and  7 


96 


ALTERNATING-CURRENT   MACHINES. 


kilogausses.  The  necessary  cross-section,  A,  of  iron,  neces- 
sary to  give  the  desired  counter  E.M.F.,  as  well  as  the 
number  of  turns  of  wire  in  the  primary,  is  then  found  from 
the  above,  as  3>m=&mA. 

(b)  With  secondary  closed  through  an  outside  impedance. 
The  flux,  which  is  linked  with  the  primary,  is  also  linked 
with  the  secondary.  Its  variations  produce  in  the  secon- 


Z 


Lighting  Transformers 


0  2  4  6  8  10         12         14          16         18      20         22         24         28 

Capacity  in  Kilowatts 
Fig.  75- 

dary  an  E.M.F.  T  times  as  great  as  the  counter  E.M.F.  in 
the  primary,  since  there  are  r  times  as  many  turns  in  the 
secondary  coil  as  in  the  primary,  or 

Es=rEp. 

If  this  secondary  be  closed  through  an  external  impedance, 
a  current  7,  will  flow  through  this  circuit.  In  the  secon- 
dary coil  the  ampere  turns,  njs,  will  be  opposed  to  the 
ampere  turns  of  the  primary,  and  will  thus  tend  to  demag- 
netize the  core.  This  tendency  is  opposed  by  a  read- 
justment of  the  conditions  in  the  primary  circuit.  Any 
demagnetization  tends  to  lessen  the  counter  E.M.F.  in  the 
primary  coil,  which  immediately  allows  more  current  to 


THE   TRANSFORMER.  97 

flow  in  the  primary,  and  thus  restores  the  magnetization  to 
a  value  but  slightly  less  than  the  value  on  open-circuited 
secondary.  Thus  the  core  flux  remains  practically  con- 
stant whether  the  secondary  be  loaded  or  not,  the  ampere 
turns  of  the  secondary  being  opposed  by  a  but  slightly 
greater  number  of  ampere  turns  in  the  primary.  So 

njt  =  nplp,  very  nearly, 

and  lt=^fp  =  Lfp. 

fig 

The  counter  E.M.F.  in  the  primary  of  a  transformer 
accommodates  itself  to  variations  of  load  on  the  secondary 
in  a  manner  similar  to  the  variation  of  the  counter  E.M.F. 
of  a  shunt  wound  motor  under  varying  mechanical  loads. 

If  the  secondary  load  be  inductive  or  condensive,  then  I8 
will  lag  or  lead  Es  by  the  same  angle  that  Ip  lags  or 
leads  Ept  still  neglecting  Rp,  Rs,  6\,  and  hysteresis.  In 
such  case  Ip  is  1 80°  from,  or  opposite  to,  7S,  and  Ep  is  oppo- 
site to  Es.  For  a  more  exact  statement  than  the  above, 
see  §  54. 

47.  Equivalent  Resistance  and  Reactance  of  a  Trans- 
former. —  If  a  current  of  definite  magnitude  and  lag  be 
taken  from  the  secondary  of  a  transformer,  a  current  of 
the  same  lag  and  r  times  that  magnitude  will  flow  in  the 
primary,  neglecting  resistance,  reluctance,  and  hysteresis. 
An  impedance  which,  placed  across  the  primary  mains, 
would  allow  an  exactly  similar  current  to  flow  as  this 
primary  current,  is  called  an  equivalent  impedance,  and  its 
components  are  called  equivalent  resistance  and  equivalent 
reactance. 

If  the  whole  secondary  circuit  of  a  transformer  with  its 
load  have  a  resistance  Rs  and  a  reactance  Xsy  and  if  the 


98  ALTERNATING-CURRENT    MACHINES. 

primary  pressure   be  Ep  and  the  secondary  total  pressure 
Ef,  then  the  current  that  will  flow  in  the  secondary  circuit  is 

E. 


•yr 

and  it  lags  behind  Et  by  an  angle  <£,  whose  tangent  is  —  • 

Therefore  V  J?.2  +  X?  =  ^ . 

*• 

If  the  equivalent  impedance  have  a  resistance  R  and  a 

reactance  X  then  the  ratios  --  and  '---  must  be  equal,  since 

K          Kt 

the  angle  of  current  lag  is  the  same  in  both  primary  and 
secondary.  And  since  the  current  in  the  equivalent  im- 
pedance has  the  same  magnitude  as  that  in  the  primary 


and 


But  Ip  =  T/., 

and  Ef  =  ~  Et, 

E. 

r  __  -  ;  ___ 

therefore, 

But 


Solving  Jt  =  -g 


which  are  the  values   of  the  equivalent  resistance  and  re- 
actance respectively. 


THE   TRANSFORMER.  99 

48.  Transformer  Losses.  —  The  transformer  as  thus  far 
discussed  would  have  100%  efficiency,  no  power  whatever 
being  consumed    in  the  apparatus.      The  efficiencies   of 
loaded  commercial  transformers  are  very  high,  being  gen- 
erally  above  95%  and  frequently  above  98%.      The  losses 
in  the  apparatus  are  due  to  (a)  the  resistance  of  the  elec- 
tric circuits,    (b)-!  reluctance  of   the    magnetic  circuit,  (c] 
hysteresis,  and  (d)  eddy  currents.     These  losses  may  be 
divided  into  core  losses  and  copper  losses,  according  as  to 
whether  they  occur  in  the  iron  or  the  wire  of  the   trans- 
former. 

49.  Core  Losses.  —  (a)  Eddy  current  loss,     if  the  core 
of  a  transformer  were  made  of  solid  iron,  strong  eddy  cur- 
rents would  be  induced  in  it.     These  currents  would  not 
only  cause  excessive  heating  of  the  core,  but  would  tend 
to  demagnetize  it,  and  would  require  excessive  currents  to 
flow  in  the  primary  winding  in  order  to  set  up  sufficient 
counter  E.M.F. 

To  a  great  extent  these  troubles  are  prevented  by  mak- 
ing the  core  of  laminated  iron,  the  laminae  being  trans- 
verse to  the  direction  of  flow  of  the  eddy  currents  but 
longitudinal  with  the  magnetic  flux.  Each  lamina  is  more 
or  less  thoroughly  insulated  from  its  neighbors  by  the 
natural  oxide  on  the  surface  or  by  Japan  lacquer.  The 
eddy  current  loss  is  practically  independent  of  the  load. 

The  E.M.F.  producing  these  eddy  currents  is  in  phase 
with  the  counter  E.M.F.  of  the  primary  coil,  both  being 

produced  by  the  same  flux.     Its  value  Ee  is  expressed  by 

jp 
the  fraction—^,  where  Pe  is  the  power  loss  in  watts  due 

*i 
to  eddy  currents,  and  ^  is  the  exciting  or  no-load  primary 

current.     The  value  of  Pe  is  calculated  from  the  following 


100  ALTERNATING-CURRENT   MACHINES. 

empirical  formula,  in  which  perfect  insulation  between  the 
laminae  is  assumed  : 


where 

k  =  a  constant  depending  upon   the   reluctivity  and 

resistivity  of  the  iron. 
v  =  volume  of  iron  in  cm.3, 
/  =  thickness  of  one  lamina  in  cm., 
f  =•  frequency, 
and 

(&,„=  maximum  flux  density  (<I>TO  per  cm.2). 

In  practice  k  has  a  value  of  about  1.6  x  io~~n. 

(b)  Hysteresis  loss.  A  certain  amount  of  power,  Ph, 
due  to  the  presence  of  hysteresis,  is  required  to  carry  the 
iron  through  its  cyclic  changes.  The  value  of  Ph  can  be 
calculated  from  the  formula  expressing  Steinmetz's  Law, 


where 

v  =  volume  of  iron  in  cm.3, 

f  =  frequency, 

($>m=  the  maximum  flux  density, 
and  rj  =  the  hysteretic  constant  (.002  to  .003). 

The   portion    of    the    impressed   E.M.F.    which    must    be 
expended  in  the  primary  circuit  to  balance  the  hysteretic 

loss  is 

j?         h 

E*  r 

This  is  in  phase  with  7r 

Closely  associated  with  Eh  is  another  portion  of  the 
impressed  E.M.F.  which  is  consumed  in  producing  the 
cyclical  and  sinusoidal  variations  of  magnetic  flux.  This  is 
not  easily  considered  distinct  from  Eh.  Consider,  however, 
the  primary  current.  There  is  but  one  primary  current. 


THE  TRANSFORMER.  IOI 

At  any  instant  of  time  a  portion  of  it  is  balanced  and  its 
magnetic  effect  is  neutralized  by  the  demagnetizing  cur- 
rent in  the  secondary ;  another  portion  is  balanced  by  the 
demagnetizing  action  of  the  eddy  currents ;  and  the  rem- 
nant is  useful  in  producing  the  cyclical  variations  of  the 
magnetic  flux.  If  the  flux  be  sinusoidal  this  portion  of 
the  current  cannot  be  sinusoidal.  This  is  due  to  the 
change  in  permeability  with  saturation  of  the  iron  core. 
Neither  is  the  rising  current  curve  the  reverse  of  the  fall- 
ing current  curve.  This  is  due  to  the  fact  that,  owing  to 
hysteresis,  the  permeability  on  rising  flux  is  smaller  than 
on  falling  under  a  given  magnetomotive  force.  This  last 
portion  of  the  primary  current  is  therefore  not  sinusoidal. 
As  it  is  but  a  small  percentage  of  the  total  current,  it  is, 
however,  for  convenience  generally  considered  as  sinusoi- 
dal. To  send  this  distorted  portion  of  the  primary  current 
requires  a  portion  of  the  impressed  E.M.F.,  and  this  is 
made  up  of  two  components,  —  Eh  in  phase  with  the  pri- 
mary current  and  discussed  above,  and  Emag  at  right  angles 
with  the  primary  current.  This  Emag  may  be  considered  as 
sending  that  portion  of  the  current  sufficient  to  overcome 
the  magnetic  reluctance  of  the  core.  Being  at  right  angles 
with  Ip  it  represents  no  loss  of  power.  During  half  of  the 
time  Ip  and  Emag  have  the  same  direction  and  during  the 
other  half  they  are  in  opposite  directions.  The  core  there- 
fore alternately  receives  energy  from  the  circuit  and  gives 
it  back  to  the  circuit. 

To  determine  the  value  of  Emag  consider  that  it  must  be 
of  such  a  magnitude  as  will  send  through  the  primary  coil, 
of  resistance  Rp,  that  portion,  Imag,  of  the  main  current 
which  produces  the  flux, 

F      —  J?  f 

•^ —  -^^^ 


102  ALTERNATING-CURRENT   MACHINES. 

Representing  the  reluctance  of  the  core  by  (R,  and  the 
magnetomotive  force  necessary  to  produce  the  flux  3>m  by 
3C,  from  §§  21  and  25,  vol.  i., 


3C 


whence  fmay  = 

4  7T  II p  4  V2   7T/^, 

and  •#*«,=  T  ?    -L     '"  • 

Ima;i  is  called  the  magnetizing  current  of  a  transformer. 
The  primary  counter  E.M.F.,  E,  is  less  than  the  primary 
line  voltage  by  the  slight  pressure  necessary  to  send  this 
current  through  the  primary  resistance,  thus, 

The  value  of  (R  is  calculated  (§  24,  vol.  i.)  from 


where  /  is  the  length  of  magnetic  circuit,  A  its  cross-sec- 
tion and  p  the  reluctivity  of  the  iron 

\  /A       permeability/ 

In  modern  commercial  transformers  the  core  loss  at 
60  ~  may  be  about  70%  hysteresis  and  30%  eddy  current 
loss.  At  1 25^ it  may  be  about  55%  hysteresis  and  45% 
eddy  current  loss.  This  might  be  expected,  since  it  was 
shown  that  the  first  power  of /enters  into  the  formula  for 
hysteresis  loss,  while  the  second  power  of/  enters  into  the 
formula  for  eddy  current  loss. 


THE   TRANSFORMER.  103 

The  core  loss  is  also  dependent  upon  the  wave-form  of 
the  impressed  E.M.F.,  a  peaked  wave  giving  a  somewhat 
lower  core  loss  than  a  flat  wave.  It  is  not  uncommon  to 
find  alternators  giving  waves  so  peaked  that  transformers 
tested  by  current  from  them  show  from  5%  to  10%  less 
core  loss  than  they  would  if  tested  by  a  true  sine  wave. 
On  the  other  hand  generators  sometimes  give  waves  so 
flat  that  the  core  loss  will  be  greater  than  that  obtained  by 
the  use  of  the  sine  wave. 

The  magnitude  of  the  core  loss  depends  also  upon  the 
temperature  of  the  iron.  Both  the  hysteresis  and  eddy  cur- 
rent losses  decrease  slightly  as  the  temperature  of  the  iron 
increases.  In  commercial  transformers,  a  rise  in  tempera- 
ture of  40°  C.  will  decrease  the  core  loss  from  5%  to  10%. 
An  accurate  statement  of  the  core  loss  thus  requires  that 
the  conditions  of  temperature  and  wave-shape  be  specified. 

The  core  loss  is  practically  constant  at  all  loads,  and  is 
the  same  whether  measured  from  the  high-tension  or  the 
low-tension  side,  the  exciting  current  in  either  case  being 
the  same  percentage  of  the  corresponding  full-load  current. 
The  exciting  current  varies  in  magnitude  with  the  design 
of  the  transformer.  In  general  it  will  not  exceed  5  %  of 
the  full-load  current,  and  in  standard  lighting  transformers 
it  may  be  as  low  as  i%.  In  transformers  designed  with 
joints  in  the  magnetic  circuit  the  exciting  current  is  largely 
influenced  by  the  character  of  the  joints,  increasing  if  the 
joint  is  poorly  constructed.  In  the  measurement  of  core 
loss,  if  the  product  of  the  impressed  volts  by  the  exciting 
current  is  less  than  twice  the  measured  watts  (i.e.,  if 
cos  <£  >.5  or  <j>  <  60°)  there  is  reason  to  suspect  poorly 
constructed  magnetic  joints  or  higher  densities  in  the  iron 
than  good  practice  allows. 


104  ALTERNATING-CURRENT   MACHINES. 

50.  Copper  Losses.  —  The  copper  losses  in  a  transformer 
are  almost  solely  due  to  the  regular  current  flowing 
through  the  coils.  Eddy  currents  in  the  conductor  are 
either  negligible  or  considered  together  with  the  eddy  cur- 
rents in  the  core. 

When  the  transformer  has  its  secondary  open-circuited 
the  copper  loss  is  merely  that  due  to  the  exciting  current 
in  the  primary  coil,  I2magRp.  This  is  very  small,  much 
smaller  than  the  core  loss,  for  both  Imag  and  Rp  are  small 
quantities.  When  the  transformer  is  regularly  loaded  the 
copper  loss  in  watts  may  be  expressed 


At  full  load  this  loss  will  considerably  exceed  the  core 
loss.  While  the  core  loss  is  constant  at  all  loads,  the 
copper  loss  varies  as  the  square  of  the  load. 

51.  Efficiency.  —  Since  the  efficiency  of  induction  appa- 
ratus depends  upon  the  wave-shape  of  E.M.F.,  it  should  be 
referred  to  a  sine  wave  of  E.M.F.,  except  where  expressly 
specified  otherwise.  The  efficiency  should  be  measured 
with  non-inductive  load,  and  at  rated  frequency,  except 
where  expressly  specified  otherwise. 

The  efficiency  of  a  transformer  is  expressed  by  the  ratio 
of  the  net  power  output  to  the  gross  power  input  or  by 
the  ratio  of  the  power  output  to  the  power  output  plus  all 
the  losses.  The  efficiency,  c,  may  then  be  written, 


where  Vt  is  the  difference  of  potential  at  the  secondary 
terminals. 

If  the  transformer  be  artificially  cooled,  as  many  of  the 


THE   TRANSFORMER.  10$* 

larger  ones  are,  then  to  this  denominator  must  be  added 
the  power  required  by  the  cooling  device,  as  power  con- 
sumed by  the  blower  in  air-blast  transformers,  and  power 
consumed  by  the  motor-driven  pumps  in  oil  or  water 
cooled  transformers.  Where  the  same  cooling  apparatus 
supplies  a  number  of  transformers  or  is  installed  to  supply 
future  additions,  allowance  should  be  made  therefor. 

Inasmuch  as  the  losses  in  a  transformer  are  affected  by 
the  temperature,  the  efficiency  can  be  accurately  specified 
only  by  reference  to  some  definite  temperature,  such  as 
25°  C. 

The  all-day  efficiency  of  a  transformer  is  the  ratio  of 
energy  output  to  the  energy  input  during  the  twenty-four 
hours.  The  usual  conditions  of  practice  will  be  met  if  the 
calculation  is  based  on  the  assumption  of  five  hours  full- 
load  and  nineteen  hours  no-load  in  transformers  used  for 
ordinary  lighting  service.  With  a  given  limit  to  the  first 
cost,  the  losses  should  be  so  adjusted  as  to  give  a  maximum 
all-day  efficiency.  For  instance,  a  transformer  supplying 
a  private  residence  with  light  will  be  loaded  but  a  few 
hours  each  night.  It  should  have  relatively  much  copper 
and  little  iron.  This  will  make  the  core  losses,  which  con- 
tinue through  the  twenty-four  hours,  small,  and  the  copper 
losses,  which  last  but  a  few  hours,  comparatively  large. 
Too  much  copper  in  a  transformer,  however,  results  in  bad 
regulation.  In  the  case  of  a  transformer  working  all  the 
time  under  load,  there  should  be  a  greater  proportion  of 
iron,  thus  requiring  less  copper  and  giving  less  copper  loss. 
This  is  desirable  in  that  a  loaded  transformer  has  usually 
a  much  greater  copper  loss  than  core  loss,  and  a  halving 
of  the  former  is  profitably  purchased  even  at  the  expense 
of  doubling  the  latter. 


IOG  ALTERNATING-CURRENT    MACHINES. 

52.  Regulation The  definition  of  the  regulation  of  a 

transformer  as  authorized  by  the  American  Institute  of 
Electrical  Engineers  is  as  follows  :  "  In  transformers  the 
regulation  is  the  ratio  of  the  rise  of  secondary  terminal 
voltage  from  full-load  to  no-load  (at  constant  impressed 
primary  terminal  voltage)  to  the  secondary  full-load  volt- 
age." Further  conditions  are  that  the  frequency  be  kept 
constant,  that  the  wave  of  impressed  E.M.F.  be  sinusoidal, 
and  that  the  load  be  non-inductive. 

Not  the  whole  primary  impressed  pressure  is  operative 
in  producing  secondary  pressure,  for  Ip  Rp  volts  are  lost  in 
overcoming  the  resistance  of  the  primary  coil.  Besides 
this  there  is  a  flux  linked  with  the  primary  that  does  not 
link  the  secondary.  This  induces  a  counter  pressure  in 
the  primary  which  neutralizes  a  part  of  the  impressed 
pressure.  Such  flux,  linking  one  coil  but  not  the  other,  is 
called  leakage  flux.  Furthermore,  not  all  of  the  E.M.F. 
induced  in  the  secondary  is  utilizable  at  the  terminals. 
There  is  a  drop  of  Is  Rs  volts  due  to  the  resistance  of  the 
secondary  coil,  and  another  drop  due  to  a  leakage  flux 
which  links  the  secondary  but  not  the  primary.  All  these 
drops  increase  with  load,  and  therefore,  neglecting  core 
loss  effects,  at  no  load  Es=rEp)  but  on  load,  E9<?EV,  and 
the  percentage  of  the  full-load  secondary  pressure  repre- 
sented by  this  fall  is  the  regulation. 

The  leakage  flux  affects  the  action  of  a  transformer  just 
the  same  as  would  an  inductance  connected  in  series  with 
the  same  transformer,  the  latter  having  no  leakage  flux. 

The  leakage  flux  increases  with  the  current ;  and  if,  for  a 
current  /  it  be  $,  then  the  value  of  the  inductance,  L,  is 

n& 

L  =      a   i 
io8/ 


THE   TRANSFORMER.  107 

where  ;/  is  the  number  of  turns  in  the  coil.  A  method  of 
calculating  L,  the  equivalent  inductance,  is  given  in  the 
next  article. 

The  resistance  of  the  secondary  causes  a  drop  fs  Ra. 
The  same  effect  on  the  regulation  would  be  caused  if  the 
secondary  resistance  were  zero  and  another  resistance 

r> 

whose  value  is  R^=  —j-  were  inserted  in  the  primary  cir- 
cuit. The  imaginary  primary  drop,  resulting  from  this 
insertion,  has  to  be  but  -  as  great  as  the  actual  secondary 

drop  to  be  as  great  a  percentage  of  the  impressed  E, 
and  there  is  r  times  as  much  current  to  cause  it,  hence 

D 

Rs  =  -°,.  The  power  lost  in  this  imaginary  resistance  is 
I*R#  and  this  equals  the  power  really  lost  in  the  secondary 
I~R,,  since  Ip  =  T/,,  and  R^  =  — ' . 

T 

In  order  to  calculate  the  regulation,  consider  this  equiv- 
alent of  secondary  drop  to  be  accounted  for  in.  the  primary. 
Then  for  a  given  impressed  E.M.F.  on  the  pr.i 
terminal  voltage  on- the  secondary  will -be 
at  no  load  £t  =  Tj^ 

at  primary  load  /JtJ 


r  <KS)  cos  <£  —  uLip  sin  <£j, 

where      Et  —  secondary  pressure  generated, 

Vt  =  difference  of  potential  at  secondary  terminals, 
L  —  Lp-\-  Lz  as  calculated  in  the  next  section, 

and  <f>  —  angle  of  lag  of  Ip  behind  Ef. 

Then  from  the  definition  of  regulation,  when  /  in  the 
above  is  made  equal  to  the  full-load  current, 

rE  —  V 
Regulation  =  — ^- — -  • 


io8 


ALTERNATING-CURRENT   MACHINES. 


53.   Calculation  of   Equivalent  Leakage   Inductance — 

The  arrangement  of  one  of  the  most  usual  kinds  of  core 

type  transformers  called 
the  "  type  H,"  is  shown 
in  Fig.  76.  The  coarse 
wire  is  wound  inside  the 
fine  wire,  and  as  these 
are  more  generally  used 
as  step-down  transformers 
the  latter  will  be  called 
the  primary. 

Fig.  77  shows  one  leg 
of  the  transformer,  giving 
the  paths  of  leakage  flux 
Fig-  76.  and  the  system  of  nota- 

tion employed.  The  discussion  is  carried  on  entirely  in 
c.  G.  s.  units.  Consider  the  secondary  (coarse  wire)  coil 
first. 


c 
2 

N 

Nvi 

g 

Primary. 

: 

L 

H 
H 

If 
ii* 

Secondary.  | 

1 

£ 

l 

SecojTCJary^_ 

i 

ii 

t 

T  - 

J 

- 

J           \__ 

Fig.  77- 

The  M.M.F.  tending  to  send  flux  through  the  elemen- 
tary portion  dx  and   back  through  the  iron  is  —  of   the 
whole  M.M.F.  of  the  secondary,  so  for  any  element, 
M.M.F.  --  '  * 


THE    TRANSFORMER.  109 

Since  the  permeability  of  iron  is  roughly  1000  times 
that  of  air,  no  appreciable  error  is  introduced  by  consider- 
ing the  whole  reluctance  of  the  circuit  of  the  leakage  flux 
to  be  in  the  air  portion  of  that  circuit.  If  it  be  assumed 
that  the  lines  of  force  follow  a  circular  path  from  the  end 
of  the  coil  to  the  iron,  the  length  of  the  air  portion  of  the 
magnetic  circuit  for  any  element  is  C  +  ir.v.  The  use  of 
this  value  will  result  in  an  integral  expression,  simple 
enough  in  theory,  but  too  unwieldy  to  be  introduced  on 
these  pages.  Since  the  portion  of  the  air  path  outside 
the  coil  (the  curved  portion)  is  a  small  part  of  the  whole 
path,  no  serious  error  will  be  'introduced  by  assuming 
that  the  leakage  flux  from  any  element  follows  a  path 

V 

whose  length  is  the  average  length  C  4-  TT  —  •     The   cross- 

section  area  of  the  air  part  of  the  magnetic  circuit  for  any 
element  is  (2  A  +28  4-  8;r)  dx.  Therefore  the  reluctance 
of  any  element  is 


/n   _ 

~ 


2  (A  +  B  +  4  x)  dx 
The  elementary  leakage  flux,  d$,  is  then 


_  M.M.F.   _  4  TT  ;/,,/>        2  (A  +  B  +  4  x)  dx 

—         '"^n  v 

' 


Since  this  flux  links  with  -—  of  the  secondary  turns,  the 

.A 

number  of  linkaes  is 


8  irnsis  (A+B  +  4  #)  xdx      xng  _  8  TT  n*  i,  (A+B  +  ^x)  xrdx 

+  -x\x  x  (c+-x\x* 

2         J  \  2         I 


110  ALTERNATING-CURRENT    MACHINES. 

By  definition  (§  8)  the  coefficient  of  self-induction,  /, 
is  numerically  equal  to  the  number  of  linkages  per  unit 
current.  Therefore 

_  linkages  _  8  TTH?(A  +  B  +  4  x)  x*dx 


The  limits  of  the  variable  x  are  o  and  X,  therefore 

?.=  ,       8V"'\     -\(A  +  E)  f**<t*  +  4  f** 
rr-_j_^  vVysi.  JQ  JQ 


This  applies  to  one  leg  of  the  transformer.      For  the  two 
legs,  upon  reverting  to  practical  units, 


io 


all  the  terms  of  which  are  either  absolute  numbers  or 
linear  dimensions  in  centimeters. 

It  cannot  be  objected  that  this  analysis  does  not  take 
account  of  the  leakage  flux  that  does  not  travel  the  whole 
length  of  the  coil,  C.  It  is  a  true  statement  for  any 
length,  and  therefore  might  be  applied  to  the  elementary 
length  dC,  which  when  integrated  would  give  the  result 
stated  above. 

The  value  of  Lr  is  determined  in  the  same  way,  and  the 


THE   TRANSFORMER.  Ill 

expression  therefore  is  quite  similar.  There  can  be  no 
iron  in  the  path  of  the  leakage  flux  from  the  outside  coil, 
so  the  reluctance  will  be  twice  as  great.  The  value  that  is 
represented  by  A  for  the  inner  coil  becomes  A  +  2X +  2g 
for  the  outer.  Likewise  B  is  replaced  by  B  -\-  2X  +  2g,  g 
being  the  space  occupied  by  insulation  between  the  coils. 
Then 

_  8  ,       M+.2^+2£-)  +  (^  +  2^-h2£)+3F 

L*-—**>V*  3(C  +  7rF) 

If  the  secondary  circuit  is  open  the  secondary  coil  is 
idle,  equivalent  to  so  much  air,  and  all  the  flux  set  up  by 
the  primary  is  leakage  flux. 

As  the   secondary  resistance   can    be    replaced    by   an 

r> 

equivalent  primary  resistance,  R^  =  — * ,  for  purposes  of 
calculation,  so  also  the  secondary  inductance  can  be  re- 
placed by  an  equivalent  inductance  in  the  primary,  L3  =  —'. 

These  values,  Lp  and  Z3,  are  to  be  used  in  the  formula  at 
the  end  of  the  last  section  for  determining  the  regulation 
of  a  transformer. 

54.  Exact  Solution  of  a  Transformer In  the  treat- 
ment of  regulation,  efficiency,  etc.,  heretofore,  certain 
small  errors  have  been  allowed,  due  to  neglecting  the 
effects  of  the  core,  eddy  currents,  and  hysteresis  losses. 
The  following  graphic  solution,  adapted  from  Steinmetz, 
takes  account  of  all  these  effects,  and  is  general  in  all 
respects. 

It  must  first  be  understood  that  there  are  three  fluxes 
to  be  considered :  (i)  The  useful  flux  that  links  both  coils. 
It  is  not  in  any  definite  phase  with  either  lp  or  7S.  It  is, 
however,  always  at  right  angles  to  the  E.M.F.  it  induces, 


112  ALTERNATING-CURRENT   MACHINES. 

the  direct  in  the  secondary,  and  the  counter  in  the  primary. 
(2)  The  leakage  flux  of  the  primary  coil.  This  links  the 
primary  only,  and  being  independent  of  fs  is  always  in 
phase  with  Ip.  (3)  The  leakage  flux  of  the  secondary  coil. 
This  is  similarly  in  phase  with  /,. 

Let       Es  =  E.M.F.  induced  in  secondary, 

Vs  =  difference    of    potential    at    secondary    ter- 
minals, 
Ep  =  impressed  primary  pressure, 

E0  =  operative  part  of  Ep  I E0  =  —  J  , 

Ip  and  Ia  =  primary  and  secondary  currents  respectively, 
<£p  and  (f>s  =  lag  of  primary   and  of  secondary  currents 

respectively  behind  Ep  and  £„ 
y  =  angle  of  lag  of  M.M.F.  behind  useful  flux. 

The  problem  is  :  Given  the  necessary  data  of  the  trans- 
former, to  determine  its  behavior  with  any  specified  load 
on  the  secondary. 

As  in  Fig.  78,  draw  the  line  3>,  representing  the  direc- 
tion of  the  flux,  vertically  for  convenience.  In  this  analy- 
sis, the  no-load  exciting  current  is  separated  into  two  com- 
ponents. One  is  used  in  neutralizing  the  demagnetizing 
effect  of  the  eddy  currents.  The  other,  fh,  is  the  magne- 
tizing current  and  is  also  made  up  of  two  components,  one 
in  phase  with  the  primary  pressure  Ep  and  the  other  at 
right  angles  with  it.  The  relative  magnitudes  of  these 
two  components  are  dependent  upon  the  shape  of  the 
hysteresis  curve  of  the  iron.  Once  determined  they  may 
be  represented  as  Ih  cos  (3  and  Ih  sin  /?  where  (3  is  termed 
the  angle  of  hysteretic  lag.  When  multiplied  by  Ep  the 
first  represents  the  power  lost  in  hysteresis  ;  the  second 
the  power  passing  backward  and  forward  between  the 


THE   TRANSFORMER. 


magnetic  field  and  the  circuit.  If  to  the  former  the  power 
lost  in  eddy  currents,  We,  be  added  and  the  two  be  com- 
bined with  the  latter  as  in  Fig.  79  an  angle  y  results,  which 


Fig.  78. 

represents  the  lag  of  the  magnetomotive  force.  Determine 
the  angle  y  in  this  manner.  Draw  the  line  M.M.F.  (Fig. 
78)  y  ahead  of  <E>  indicating  in  direction  and  magnitude  the 
ampere  turns  which  must  exist  to  set  up 
the  flux  <E>.  Its  value  is  determined  dur- 
ing the  transformer  design.  Draw  from 
the  center  the  line  Em  90°  ahead  of  the 
flux,  representing  the  operative  primary 

pressure.     Its   length    is  -  Et   and  as  it 

opposes  the  counter  primary  pressure,  it  is 
set  ahead  of  the  flux.     Draw  the  line  E# 
representing  the  pressure   induced  in  the 


We     1~E  pi  7l  cos/3 
Fig.  79- 

90°  behind  3>, 


secondary.     Its  length  is  proportional  to  the  no-load  sec- 
ondary terminal  pressure. 


114  ALTERNATING-CURRENT    MACHINES. 

The  angle  <f>s,  the  lag  due  to  the  whole  secondary  cir- 
cuit, is  known.  Draw  I8  at  <f>s  behind  ES)  and  extend  it 
till  its  length  is  proportional  to  the  secondary  ampere 
turns,  /.  nt.  This  line  represents  one  component  of  the 
magnetizing  force.  From  this  component  line  and  the 
resultant  line  M.M.F.  determine  the  other  component 
Ip  np.  Divide  this  by  np  and  the  primary  current  is  dis- 
covered in  magnitude  and  phase. 

There  is  a  drop  of  Ip  Rp  volts  in  the  primary.  The  im- 
pressed pressure  that  compensates  for  this  is  in  phase  with 
fp.  A  counter  voltage  90°  behind  Ip  will  be  set  up  due  to 
the  primary  leakage  flux.  Its  value  is  &LP  Ip.  To  over- 
come this  an  impressed  pressure  must  be  supplied  opposite, 
it  in  phase  or  90°  ahead  of  the  current.  In  a  side  figure 
vectorially  add  IpRp  in  the  phase  of  Ip  and  <»LPIP  90° 
ahead  of  this  phase.  This  gives  the  direction  and  magni- 
tude of  the  drop  dp  in  the  primary.  Properly  add  dp  to  the 
operative  pressure  E0  and  the  necessary  impressed  pressure 
Ep  is  the  resultant.  The  angle  between  Ip  and  Ep  is 
the  angle  of  lag  <j>p  of  the  primary  current.  It  slightly 
exceeds  <£.. 

The  pressure  £.  is  generated  in  the  secondary  coil. 
There  is  a  drop  of  ft  R9  volts  in  this  coil  in  phase  with  /,. 
A  counter  voltage  90°  behind  7S  will  be  set  up  due  to  the 
secondary  leakage  flux.  Its  value  is  «>LS  It.  To  overcome 
this  <oZa  I%  volts  generated  at  180°  from  this  (i.e.,  90°  ahead 
of  /,)  will  be  consumed.  In  a  side  figure  vectorially  add 
/.  Rt  in  the  phase  of  It  and  ^Ltlt  at  90°  ahead  of  this 
phase.  This  gives  the  drop  dt  in  the  secondary  coil.  This 
drop  must  be  subtracted  from  the  pressure  generated  to 
give  the  secondary  terminal  volts.  To  subtract  a  vector, 
revolve  it  180°  and  proceed  as  in  addition.  Properly  sub- 


THE   TRANSFORMER. 


tract  dt  from  Et  and  the  resultant,  Vt,  is  the  potential  dif- 
ference at  the  terminals  of  the  secondary  coil  of  the 
transformer. 

By  constructing  this  diagram  for  full  load  7S  and  then 
for  It  =  o,  the  regulation  of  a  transformer  can  be  found  by 
the  ratio  of  the  difference  between  the  values  of  Vt  in  each 
case  to  the  full  load  Vt.  The  efficiency  at  any  load  can  be 
determined  from  the  diagram  for  that  load,  by 

EJS  cos  <fc 


to* 


Fig.  78  is  not  the  true  diagram  of  a  commercial  trans- 
former. For  clearness  a  ratio  of  I  to  I  has  been  portrayed 
and  the  losses  greatly  exaggerated.  In  practice  it  will  be 
found  impossible  to  complete  the  solution  graphically 
because  of  the  extreme  flatness  of  the  triangles.  The 
better  way  is  to  draw  an  exaggerated  but  clear  diagram, 
and  obtain  the  true  values  of  the  sides  by  the  algebra  of 
complex  imaginary  quantities,  or  if  the  student  is  unfa- 
miliar with  this  method,  by  the  more  laborious  methods  of 
trigonometry  and  geometry. 

55.  Methods  of  Connecting  Transformers  --  There  are 
numerous  methods  of  connecting  transformers  to  distribut- 
ing circuits.  The  simplest  case  is 
that  of  a  single  transformer  in  a 
single-phase  circuit.  Fig.  80  shows 
such  an  arrangement.  This  and 
the  succeeding  figures  have  the 
pressure  and  current  values  of  the 
different  parts  marked  on  them,  as- 
suming in  each  case  a  i  K.W.,  i  to  10 
step-down  transformer.  As  in  Fig.  81,  two  or  more  trans- 


Fig> 


n6 


ALTERNATING-CURRENT    MACHINES. 


formers  may  have  their  primaries  in  parallel  on  the  same 
circuit,  and  have  their  secondaries  independent.  If  the  two 
secondaries  of  this  case  are  connected  properly  in  series 
a  secondary  system  of  double  the  potential  will  result,  or 
by  adding  a  third  wire  to  the  point  of  juncture,  as  shown 
by  the  dotted  line  of  Fig.  82,  a  three-wire  system  of  dis- 
tribution can  be  secured.  The  secondaries  must  be  con- 
nected cumulatively  ;  that  is,  their  instantaneous  E.M.F.'s 
must  be  in  the  same  direction.  If  connected  differentially, 
there  would  be  no  pressure  between  the  two  outside  see- 


to  A. 


Fig.  81.  Fig.  82. 

ondary  wires,  the  instantaneous  pressures  of  the  two  coils 
being  equal  and  opposed  throughout  the  cycle.  Again, 
with  the  same  condition  of  primaries,  the  secondaries  can 
be  connected  in  multiple  as  in  Fig.  83.  Here  the  connec- 
tions must  be  such  that  at  any  instant  the  E.M.FJs  of  the 
secondaries  are  toward  the  same  distributing  wire.  The 
connection  of  more  than  two  secondaries  in  series  is  not 
common,  but  where  a  complex  network  of  secondary  dis- 
tributing mains  is  fed  at  various  points  from  a  high-tension 
system,  secondaries  are  necessarily  put  in  multiple. 

In  many  types  of  modern  transformers  it  is  usual  to 


THE   TRANSFORMER. 


117 


Fig.  83. 


wind  the  secondaries  (low-tension)  in  two  separate  and 
similar  coils,  all  four  ends  being  brought  outside  of  the 
case.  This  allows  of  connections  to  two-wire  systems  of 
either  of  two  pressures,  or  for  a  three-wire  system  accord- 
ing to  Figs.  82  and  83,  to  be 
made  with  the  one  transformer, 
this  being  more  economical  than 
using  two  transformers  of  half 
the  size,  both  in  first  cost  and 
in  cost  of  operation.  In  many 
transformers  the  primary  coils 
are  also  wound  in  two  parts. 
In  these,  however,  the  four  ter- 
minals are  not  always  brought 
outside,  but  in  some  cases  are 
led  to  a  porcelain  block  on  which  are  four  screw-connectors 
and  a  pair  of  brass  links,  allowing  the  coils  to  be  arranged 
in  series  or  in  multiple  according  to  the  pressure  of  the 
line  to  which  they  are  to  be  connected.  From  this  block 

two  wires  run  through   suitably 
bushed  holes  outside  the  case. 
A  two-phase  four-wire  system 
can  be  considered  as  two   inde- 
pendent    single-phase     systems, 
transformation      being      accom- 
plished by  putting  similar  single- 
Fis-  84-  phase  transformers  in  the  circuit, 

one  on  each  phase.  If  it  is  desired  to  tap  a  two-phase 
circuit  to  supply  a  two-phase  three-wire  circuit,  the 
arrangement  of  Fig.  84  is  employed.  By  the  reverse 
connections  two-phase  three- wire  can  be  transformed  to 
two-phase  four-wire.  An  interesting  transformer  conneo 


nS 


ALTERNATING-CURRENT    MACHINES. 


tion  is  that  devised  by  Scott,  which  permits  of  transfor- 
mation from  two-phase  four-wire  to  three-phase  three-wire. 
Fig.  85  shows  the  connections  of  the  two  transformers. 
If  one  of  the  transformers  has  a  ratio  of  10  to  i  with  a 
tap  at  the  middle  point  of  its  secondary  coil,  the  other 

must  have  a  ratio  of  10  to  .867  fio  to — -1  One  ter- 
minal of  the  secondary  of  the  latter  is  connected  to  the 


Fig.  85. 


100 

Fig.  86. 


middle  of  the  former,  the  remaining  three  free  terminals 
being  connected  respectively  to  the  three-phase  wires.  In 
Fig.  86,  considering  the  secondary  coils  only,  let  mn  rep- 
resent the  pressure  generated  in  the  first  transformer. 
The  pressure  in  the  second  transformer  is  at  right  angles 
(§  5)  to  that  in  the  first,  and  because  of  the  manner  of 
connection,  proceeds  from  the  center  of  mn.  Therefore 
the  line  op  represents  in  position,  direction,  and  magnitude 
the  pressure  generated  in  the  second.  From  the  geo- 
metric conditions  mnp  is  an  equilaterial  triangle,  and  the 
pressures  represented  by  the  three  sides  are  equal  and  at 
60°  with  the  others.  This  is  suitable  for  supplying  a 
three-phase  system.  In  power  transmission  plants  it  is 
not  uncommon  to  find  the  generators  wound  two-phase, 
and  the  step-up  transformers  arranged  to  feed  a  three- 
phase  line. 


THE   TRANSFORMER. 


In  America  it  is  common  to  use  one  transformer  for 
each  phase  of  a  three-phase  circuit.  The  three  transform- 
ers may  be  connected  either  Y  or  A.  They  may  be  Y  on 
the  primary  and  A  on  the  secondary,  or  vice  versa.  Fig. 
87.  shows  both  primary  and 
secondary  connected  A. 
The  pressure  on  each  pri- 
mary is  1000  volts,  and  as 
a  I-K.W.  transformer  was 
assumed,  i.e.,  I  K.W.  per 
phase,  there  will  be  one 
ampere  in  each,  calling  for 
T-7  (^3)  amperes  in  each  Fig.  87. 

primary  main  (§  33).  This  arrangement  is  most  desirable 
where  continuity  of  service  is  requisite,  for  one  of  the 
transformers  may  be  cut  out  and  the  system  still  be 
operative,  the  remaining  transformers  each  taking  up  the 


r 

5              1TA. 

i 

I 

T 

I 

I 

.el 

17  A. 

difference  between  4  and 


the  full  load  ;  that  is,  if  the 
system  was  running  at 
full  load,  and  one  trans- 
former was  cut  out,  the 
other  two  would  be  over- 
loaded i6|  per  cent. 
Even  if  two  of  them 
were  cut  out,  service 
over  the  remaining  phase 
could  be  maintained.  It 


Fig.  88. 


ularly  supply  motors  from  three-phase  mains  by  two  some- 
what larger  transformers  rather  than  by  three  smaller 
ones.  Fig.  88  shows  the  connections  for  both  primaries 
and  secondaries  in  Y.  If  in  this  arrangement  one  trans- 


120 


ALTERNATING-CURRENT   MACHINES. 


former  be  cut  out,  one  wire  of  the  system  becomes  idle, 
and  only  a  reduced  pressure  can  be  maintained  on  the  re- 
maining phase.  The  advantage  of  the  star  connection  lies 
in  the  fact  that  each  transformer  need  be  wound  for  only 
57.7  per  cent  of  the  line  voltage.  In  high-tension  trans- 
mission this  admits  of  building  the  transformers  much 
smaller  than  would  be  necessary  if  they  were  A  connected. 
Fig.  89  shows  the  connections  for  primaries  in  A,  second- 
aries in,  Y  ;  and  Pig.  90  those  for  primaries  in  Y  and  sec- 
ondaries in  A.  By  taking  advantage  of  these  last  two 
arrangements,  it  is  possible  to  raise  or  lower  the  voltage 


Fig.  89. 


Fig.  90. 


with  i  to  i  transformers.  With  three  I  to  i  transform- 
ers, arranged  as  in  Fig.  89,  100  volts  can  be  transformed 
to  173  volts;  while  if  connected  as  in  Fig.  90,  100  volts 
can  be  transformed  into  58  volts. 

Fig.  91  shows  a  transformer  and  another  one  connected 
as  an  autotransformer  doing  the  same  work.  Since  the 
required  ratio  of  transformation  is  i  to  2,  the  autotrans- 
former does  the  work  of  the  regular  transformer  with  one- 
half  the  first  cost,  one-half  the  losses,  and  one-half  the 
drop  in  potential  (regulation).  The  only  objection  to  this 
method  of  transformation  is  that  the  primary  and  second- 


THE   TRANSFORMER. 


121 


ary  circuits  are  not  separate.  With  the  circuits  grounded 
at  certain  points,  there  is  danger  that  the  insulation  of  the 
low-tension  circuit  may  be  subjected  to  the  voltage  of  the 
high-tension  circuit.  One  coil  of  an  autotransformer  must 
be  wound  for  the  lower  voltage,  and  the  other  coil  for  the 


^JOne  100  Kw 
Transformer 
2000  v.    2: 1  Ratio  1  to  2 


Losses  not  considered 


One  50  Kw. 
Transformer 
loco  v.        o."  Ratio  1  to  1 


Losses  not  considered 
Fig.  91. 


difference  between  the  two  voltages  of  transformation. 
The  capacity  of  an  autotransformer  is  found  by  multiply- 
ing the  high-tension  current  by  the  difference  between  the 
two  operative  voltages.  Autotransformers  are  often  called 
compensators.  Compensators  are  advantageously  used 


Three  16.5  Kw. 
Transformers 
Ratio  1  to  1 


Loss.es  not  considered 


Losses  not  considere 


Fig.  92. 

where  it  is  desired  to  raise  the  potential  by  a  small 
amount,  as  in  boosting  pressure  for  very  long  feeders. 
Fig.  92  shows  three  I  to  2  transformers  connected  in  A  on 
a  three-phase  system,  and  three  I  to  i  compensators  con- 
nected in  Y  to  do  the  same  work. 

From  a  two-phase  circuit,  a  single-phase  E.M.F.  of  any 


122 


ALTERNATING-CURRENT    MACHINES. 


desired  magnitude  and  any  desired  phase-angle  may  be 
secured  by  means  of  suitable  transformers,  as  shown  in 
Fig-  93-  Suppose  the  two  phases  X and  Fof  a  two-phase 
system  be  of  100  volts  pressure,  and  it  is  desired  to  obtain 
a  single-phase  E.M.F.  of  1000  volts  and  leading  the  phase 
X  by  30°.  As  in  Fig.  94,  draw  a  line  representing  the 


Fig.  93. 


DIRECTION  OF  PHASE  X. 

Fig.  94. 


direction  of  phase  X.  At  right  angles  thereto,  draw  a  line 
representing  the  direction  of  phase  Y.  From  their  inter- 
section draw  a  line  1000  units  long,  making  an  angle  of 
30°  with  X.  It  represents  in  direction  and  in  length  the 
phase  and  the  pressure  of  the  required  E.M.F.  Resolve 
this  line  into  components  along  X  and  Y,  and  it  becomes 
evident  that  the  secondary  of  the  transformer  connected 
to  X  must  supply  the  secondary  circuit  with  866  volts 
and  that  the  secondary  of  the  other  must  supply  500  volts. 
Therefore  the  transformer  connected  to  X  must  step-up 
i  to  8.66  and  that  connected  to  Fmust  step-up  i  to  5.  If 
10  amperes  be  the  full  load  on  the  secondary  circuit,  the 
first  transformer  must  have  a  capacity  of  8.66  K.W.,  and  the 
second  a  capacity  of  5  K.W.  The  load  on  X  and  Y  is  not 
balanced. 

56.  Lighting  Transformers.  —Because  of  their  extensive 
use  on  lighting  distributing  systems,  the  various  manufac- 
turers have  to  a  great  extent  standardized  their  lines  of 
lighting  transformers.  Power  transformers  are  not  as  yet 


THE   TRANSFORMER.  123 

well  standardized,  probably  because  they  are  generally  used 
in  such  large  units  as  to  warrant  a  special  design  for 
each  case. 

The  Wagner  Electric  Mfg.  Co.'s  "type  M  "  transformer 
is  illustrated  in  Fig.  95.  It  is  of  the  shell  type  of  con- 
struction, makers  using  this  type  claiming  for  it  superiority 
of  regulation  and  oool  running.  In  the  shell  type  the  iron 


Fig.  95- 

is  cooler  than  the  rest  of  the  transformer,  in  the  core  type 
it  is  hotter.  As  the  "ageing  "  of  the  iron,  or  the  increase 
of  hysteretic  coefficient  with  time,  is  believed  to  be  aggra- 
vated by  heat,  this  is  claimed  as  a  point  of  superiority  of 
the  shell  type.  However,  the  prime  object  in  keeping  a 
transformer  cool  is  not  to  save  the  iron,  but  to  protect  the 
insulation  ;  and  as  the  core  type  has  less  iron  and  generally 
less  iron  loss,  the  advantages  do  not  seem  to  be  remarkably 


I24 


ALTERNATING-CURRENT    MACHINES. 


in  favor  of  either.      In  the  Wagner  "type  M  "  transformers 

the  usual  practice  of  having  two  sets  of  primaries  and  sec- 
ondaries is  followed. 
Fig.  96  shows  the  three 
coils  composing  one 
set.  A  low-tension 
coil  is  situated  between 
two  high-tensioned 
coils,  this  arrangement 
being  conducive  to 
good  regulation.  Th  • 
ideal  method  would  be 
to  have  the  coils  still 
more  subdivided  and 
interspersed,  but  prac- 
tical reasons  prohibit 
this.  Fig.  97  shows 
the  arrangement  of  the 

coils  in  the  shell.     The  space  between  the  coils  and  the 

iron  is  left  to  facilitate  the  circulation  of  the  oil  in  which 

they  are  submerged. 

The   laminae    for  the 

shell      are      stamped 

each  in  two  parts  and 

assembled  with  joints 

staggered.    As  can  be 

seen    from    the    first 

cut,  all  the  terminals 

of    the   two    primary 

and   the    two    secon- 


Fig.  96. 


___X^          "*v__ 

\°7 


Fig.  97. 


dary  coils  are  brought  outside  the  case.     The  smaller  sizes 
of  this  line  of    transformers,  those  under  1.5  K.W.,  have 


THE   TRANSFORMER. 


125 


sufficient  area  to  allow  their  running  without  oil,  so  the 
manufacturers  are  enabled  to  fill  the  retaining  case  with 
an  insulating  compound  which  hardens  on  cooling. 

The  General  Electric  Co.'s  "  H  "  transformers  are  of 
the  core  type.  In  Fig.  76  was  shown  a  sectional  view  giv- 
ing a  good  idea  of  the  arrangement  of  parts  in  this  type. 
Fig.  71  is  also  one  of  this  line  of  transformers.  In  it  is 
shown  the  tablet  board  of  porcelain  on  which  the  connec- 
tions of  the  two  high-tension  coils  may  be  changed  from 
series  to  parallel  or  vice  versa, 
so  that  only  two  high-tension 
wires  are  brought  through  the 
case.  Fig.  98  shows  the  ar- 
rangement of  the  various  parts 
in  the  assembled  apparatus. 
The  makers  claim  for  this  type 
that  the  coils  run  cooler  because 
of  their  being  more  thoroughly 
surrounded  with  oil  than  those 
of  the  shell  type.  Another 
point  brought  forward  is  that 
copper  is  a  better  conductor  of 
heat  than  iron  ;  the  heat  from  the  inner  portions  of  the 
apparatus  is  more  readily  dissipated  than  in  the  shell  type. 
The  core  has  the  advantage  of  being  made  up  of  simple 
rectangular  punchings,  and  the  disadvantage  of  having  four 
instead  of  two  joints  in  the  magnetic  circuit.  A  particular 
advantage  of  the  "type  H  "  transformer  is  the  ease  and 
certainty  with  which  the  primary  windings  can  be  sepa- 
rated from  the  secondary  windings.  A  properly  formed 
seamless  cylinder  of  fiber  can  be  slipped  over  the  inner 
winding  and  the  outer  one  wound  over  it.  This  is  much 


Fig.  98. 


126  ALTERNATING-CURRENT    MACHINES. 

more  secure  than  tape  or  other  material  that  has  to  be 
wound  on  the  coils. 


Fig.  99. 

The  Westinghouse  "  O.  D."  transformers  are  of  the 
shell  type.  The  construction  of  the  separate  parts  is 
shown  in  Fig.  99.  The  coils  are  wound  narrow  and  to  the 
full  depth,  and  high-tension  and 
low-tension  coils  alternate  side 
by  side  instead  of  from  the 
center  out.  Fig.  100  shows  a 
2  K.W.  O.  D.  transformer  with- 
out the  case.  A  tablet  board 
is  used  for  the  terminals  of 
the  high-tension  coils,  but  the 
low-tension  wires  are  all  run 
out  of  the  case.  Fig.  101 
shows  one  of  the  coils.  Type 
O.  D.  transformers  are  built 
from  J  to  25  K.W.  for  lighting 
and  to  50  K.W.  for  power. 


Those  of   10  K.W.  or  less  are 


Fig.  loo. 


THE   TRANSFORMER. 


127 


Fig.  101. 

in  cast-iron  cases,  those  above   10  K.W.  in  corrugated  iron 
cases  with  cast  tops  and  bottoms.     The  corrugations  quite 


The    windings 


Fig.   102. 

materially  increase  the  radiating  surface, 
are  submerged  in  oil. 

An  example  of  the  Stanley  Electric  Manufacturing  Co.'s 


123 


ALTERNATING-CURRENT   MACHINES. 


standard  line  of  "type  A.  O."  transformers  is  given  in 
Fig.  1 02.  These  are  also  of  the  shell  type,  with  divided 
primaries  and  secondaries,  four  of  the  eight  which  belong 
to  a  single  transformer  being  shown  in  Fig.  103. 


Fig.  103. 

57.   Cooling  of  Transformers The  use  of  oil  to  assist 

in  the  dissipation  of  the  heat  produced  during  the  opera- 
tions of  transformers  is  almost  universal  in  sizes  of  less 
than  about  100  K.W.,  especially  if  designed  for  outdoor 
use.  Some  small  transformers  are  designed  to  be  self- 
ventilating,  taking  air  in  at  the  bottom,  which  goes  out  at 
top  as  a  result  of  being  heated.  They  are  not  well  pro- 
tected from  the  weather,  and  are  liable  to  have  the  natural 
draft  cut  off  by  the  building  of  insects'  nests.  Larger 
transformers  that  are  air  cooled  and  that  supply  their  own 
draft  are  used  to  some  extent  in  central  stations  and  other 
places  where  they  can  be  properly  protected  and  attended 
to.  A  forced  draft  is,  however,  the  more  common.  Where 
such  transformers  are  employed,  there  are  usually  a  number 


THE   TRANSFORMER. 


I29 


of  them  ;  and  they  are  all  set  up  over  a  large  chamber  into 
which  air  is  forced  by  a  blower,  as  indicated  in  Fig.  104. 


Fig.  104. 


Dampers  regulate  the  flow  of  air  through  the  transformers. 
They  can  be  adjusted  so  that  each  transformer  gets  its 
proper  share. 

Fig.  105  shows  a  General  Electric  Company's  air-blast 
transformer  in  process  of  construction.  The  iron  core  is 
built  up  with  spaces  between  the  laminae  at  intervals  ;  and 
the  coils,  which  are  wound  very  thin,  are  assembled  in 
small  intermixed  groups  with  air  spaces  maintained  by 
pieces  of  insulation  between  them.  The  assembled  struc- 
ture is  subjected  to  heavy  pressure,  and  is  bound  together 
to  prevent  the  possibility  of  vibration  in  the  coils  due  to 
the  periodic  tendency  to  repulsion  between  the  primary  and 
the  secondary.  These  transformers  are  made  in  sizes  from 
100  K.W.  to  1000  K.W.  and  for  pressures  up  to  35,000  volts. 

Another  method  of  cooling  a  large  oil  transformer  is  to 
circulate  the  oil  by  means  of  a  pump,  passing  it  through  a 
radiator  where  it  can  dissipate  its  heat.  Again  cold  water 
is  forced  through  coils  of  pipe  in  the  transformer  case,  and 
it  takes  up  the  heat  from  the  oil.  There  is  the  slight  dan- 
ger in  this  method  that  the  pipes  may  leak  and  the  water 
may  injure  the  insulation.  Water-cooled  transformers 
have  been  built  up  to  2000  K.W.  capacity. 


130  ALTERNATING-CURRENT   MACHINES. 

In  those  cases  where  the  transformer  requires  some 
outside  power  for  the  operation  of  a  blower  or  a  pump, 
the  power  thus  used  must  be  charged  against  the  trans- 


Fig.  105. 

former  when  calculating  its  efficiency.  In  general  this 
power  will  be  considerably  less  than  i  %  of  the  trans- 
former capacity. 

58.  Constant-Current  Transformers.  —  For  operating 
series  arc-light  circuits  from  constant  potential  alternating- 
current  mains,  a  device  called  a  constant-current  trans- 
former  is  frequently  employed.  A  sketch  showing  the 
principle  of  operation  is  given  in  Fig.  106.  A  primary 
coil  is  fixed  relative  to  the  core,  while  a  secondary  coil  is 


THE   TRANSFORMER. 


allowed  room  to  move  from  a  close  contact  with  the 
primary  to  a  considerable  distance  from  it.  This  secon- 
dary coil  is  nearly  but  not  entirely 
counter-balanced.  If  no  current 
is  taken  off  the  secondary  that 
coil  rests  upon  the  primary. 
When,  however,  a  current  flows 
in  the  two  coils  there  is  a  repul- 
sion between  them.  The  counter- 
poise is  so  adjusted  that  there  is 
an  equilibrium  when  the  current  Flg>  lo6' 

is  at  the  proper  value.  If  the  current  rises  above  this 
value  the  coil  moves  farther  away,  and  there  is  an  increased 
amount  of  leakage  flux.  This  lowers  the  E.M.F.  induced 


Fig.  107. 


132 


ALTERNATING-CURRENT   MACHINES. 


in  the  secondary,  and  the  current  falls  to  its  normal  value. 
Thus  the  transformer  automatically  delivers  a  constant 
current  from  its  secondary  when  a  constant  potential  is 
impressed  on  its  primary. 

Fig.  107  shows  the  mechanism  of  such  an  apparatus  as 
made  by  the  General  Electric  Company.  The  cut  is  self- 
explanatory.  Care  is  taken  to  have  the  leads  to  the  mov- 


Fig.   108. 

ing  coil  very  flexible.  Transformers  for  50  lamps  or 
more  are  made  with  two  sets  of  coils,  one  primary  coil 
.being  at  the  bottom,  the  other  at  the  top.  The  moving 
coils  are  balanced  one  against  the  other,  avoiding  the 
necessity  of  a  very  heavy  counterweight.  Fig.  108  shows 
a  5o-light  constant-current  transformer  without  its  case. 
Fig.  109  shows  a  complete  25-lamp  apparatus.  The  tank 


THE   TRANSFORMER.  133 

is  filled  with  oil,  the  same  as  an  ordinary  transformer. 
Great  care  must  be  taken  to  keep  these  transformers  level, 
and  to  assist  in  this  the  larger  sizes  have  spirit-levels  built 


Fig.  109. 

into  the  case.  A  pair  of  these  transformers  can  be  spe- 
cially wound  and  connected  to  supply  a  series  arc-light 
circuit  from  a  three-phase  line,  keeping  a  balanced  load  on 
the  latter. 

59.  Design  of  a  Transformer The  method  of  design- 
ing a  transformer  depends  upon  the  specifications  as  to  con- 
struction and  operation,  and  upon  various  values  which 
the  designer  is  forced  or  sees  fit  to  assume.  The  following 
is  one  method  :  — 

Specifications.  —  These  usually  give  the  capacity  in 
watts,  the  frequency,  the  primary  voltage,  the  secondary 


134 


ALTERNATING-CURRENT    MACHINES. 


voltage,  and  the  conditions  of  operation,  place  of  installa- 
tion, whether  loaded  all  day  or  not,  etc. 

Assumptions.  —  The  assumption  of  the  following  quanti- 
ties is  usually  preliminary  to  any  calculation,  —  the  shape 
of  transformer,  -the  current  density  in  the  primary,  the 
current  density  in  the  secondary,  the  turns  in  the  primary 
coil,  and  the  maximum  flux  density  in  the  iron.  The 
method  of  design  is  one  of  cut  and  try.  A  number  of 
values  of  flux  density  and  various  numbers  of  primary  turns 
are  assumed.  Efficiency  curves  are  calculated  for  the 
various  arrangements.  The  most  efficient  is  ultimately 
selected ;  or  if  none  are  satisfactory,  the  course  of  the 
design  will  have  brought  out  the  proper  direction  to  take 
in  making  new  assumptions. 

The  following  design  refers  to  a  core-type,  step-down, 
lighting  transformer  of  about  5  K.  w.  capacity.  The  as- 
sumptions are:  1000  circular  mils  per  ampere  in  the 
primary,  1500  circular  mils  per  ampere  in  the  secondary 
(because  this  is  inside,  and  has  less  opportunity  of  dissi- 


Fig.  no. 


pating  its  heat),  500,  700,  and  1000  turns  primary 
successively,  and  2000,  3000,  and  4000  gausses  maximum 
flux  density.  The  transformer  will  have  the  shape  shown 
in  Fig.  1 1  o.  Because  of  the  general  use  of  the  English 
units  of  measure  by  most  practical  mechanics,  the  dimen- 


THE   TRANSFORMER.  135 

sions   indicated   are    all  expressed  in    inches.     The   ratio 

-  =  m  may  be  conveniently  assumed  as  m  =  1.5,  and  the 
b 

ratio  —  =  n  is  likewise  generally  made  n  =  i. 
a 

I.  To  obtain  the  area,   A,   of  the  core    in  square  centi- 
meters. 

Let         E  =  impressed  primary  E.M.F., 

(Bm=  assumed  maximum  flux  density, 
Tp  =  assumed  number  of  turns  in  primary, 
and  /  =  frequency. 

The  instantaneous  value  of  the  counter  E.M.F.  of  self- 
induction  will  be  (§  13,  vol.  i.,  §  3) 

>    sin  «/ 


d 

e  =  —  Tp$mu>  cos  a>/, 
em=-  2ir/TpQm, 

because  the  maximum  value  of  the  cosine  is  unity. 

e=~=-  V^TT/T^. 

V2 

At  no  load  this  is  equal  and  opposite  to  the  primary  im- 
pressed pressure,  so  remembering  that 


10° 

.-.  A  = 


II.    To  obtain  c  and  d  in  inches, 


136  ALTERNATING-CURRENT    MACHINES. 


-45 


2-54 

and  d  =  -—-> 

11  2.54 

III.     To  obtain  the  depth  of  coil  winding  tt 
inches. 

Let  dp  =  diameter  of  primary  wire,  including  insulation, 

in  inches, 

dt  =  diameter  of  secondary  wire,  including  insula- 
tion, in  inches, 

as  found  from  a  wire  table  ;  then,  allowing  \  inch  at  each 
end  for  insulation, 

T.,  , 


approximately,   since   but   half    the   primary  is   wound   on 
each  limb  and 


where  r  is  the  ratio  of  transformation,  — -. 

EP 
The  value  of  a  is  found  in  the  next  paragraph. 

IV.    To  obtain  a  and  b  in.  inches.     Evidently  the  trans- 
former could 'not  be  assembled  unless 

b  >  2  (tf  +  tt  -|-  insulation  and  clearance). 
Assume  b  =  2  (t,  +  /.  +  |J  - 


THE   TRANSFORMER.  137 

Now  a  =  mb, 

and  /,'  =  ^ 


so  a  =  2 

i' 

5 
But  also,  tp  =  - 


i 
a 

2 


so  substituting  and  transposing, 

(•'-BKH-.4-OT    - 

All  the  terms  of  the  right-hand  member  are  known,  so 
it  may  be  reduced  to  a  simple  number,  and  set  equal  to  K. 
Then 


and 


m 

V.      To  obtain  the  volume  v  of  iron  in  cubic  centimeters. 
—  About  90%  of  a  volume  occupied  by  laminated  iron  is 
metal. 


v  =  2  (a  +  b  +  2  c)  X  c  X  d  X  2.54    X  0.9. 

VI.      To  obtain  the  watts  Ph  lost  in  hysteresis.  —  Accord- 
ing to  Steinmetz's  Law,  using  r;  =  .003, 

Hysteresis  loss  =  .003  v&m1M  ergs  per  cycle. 


138  ALTERNATING-CURRENT    MACHINES. 

VII.  To  obtain  the  resistance  of  the  secondary  Rg  in 
ohms. — Although  surrounding  a  rectangular  core,  the  coils 
are  usually  approximately  circular  in  section,  for  con- 
venience in  winding  and  in  insulating.  If  the  section  of 
the  core  varies  considerably  from  the  square,  allowance 
can  be  made  in  estimating  the  length  of  a  mean  turn. 

Considering  the  coil  as  truly  cylindrical,  and  allowing  -| 
inch  insulation  between  it  and  the  core,  the  length  of  a 
mean  turn 


s+H) 


/  =  2  ..  , 

V2 

The  total  length  of  secondary  wire  (both  limbs)  is  then 
rTpl,  and  its  resistance  can  be  found  directly  in  a  wire 
table  giving  the  hot  resistances  of  wires  ;  or,  it  may  be 
assumed  that  the  transformer  will  operate  at  such  a  tem- 
perature that  one  mil  foot  has  1 1  ohms  resistance,  then 


II  rTp  2  TT 


•R»  — 


circular  mils 


VIII.      To  obtain   the   resistance  of  the  primary  Rp  in 
ohms.  —  Similarly  to  the  above,  the  length  of  a  mean  turn 

,=  ,,('    +'+,.+   3+ 

\^2  8  l6 

allowing  T3g-  inch  insulation  between  the  two  coils,  and  the 
total  length  of  primary  wire  is  Tpl. 

The  resistance  can  be  found  in  a  table,  or  calculated 
from 


R 


ii 

1.41 


p  12  X  circular  mils 


THE   TRANSFORMER.  139 

IX.  To  obtain  tJie  foucault  current  loss  Pf  in  watts.  — 
Steinmetz  has  given  the  empirical  formula 

Pf=io-*v(XfS>m)*t 

where  x  is  the  thickness  in  mils  of  one  lamina.  Trans- 
-former  iron  may  be  assumed  to  be  from  10  to  20  mils  in 
thickness. 

X.  To  obtain  the  efficiency  at  any  load,  /,,  in  per  cent.  — 

t  +  (r!$Rf  IC 

for  a  lighting  transformer.  If  the  load  be  inductive  the 
term  J5JS,  whenever  it  occurs,  must  be  multiplied  by  the 
power  factor  (cos  <j>).  The  error  involved  in  the  assump- 
tion Ip  =  rlt  is  negligible. 

After  calculating  the  values  in  each  of  the  preceding 
steps  for  the  three  values  of  Tp  and  the  three  values  of 
(&m  suggested,  the  efficiency  curve  of  each  transformer 
should  then  be  drawn,  taking  points  at  TL,i,  i,  f ,  and  full 
load.  After  having  selected  the  most  suitable,  determine 
the  following  values. 

XI.  To  determine  the  all-day  efficiency  in  per  cent.  — 
The  average  lighting  transformer  is  found  to  be  loaded 
equivalent  to  full  load  for  5   hours,  and  no  load  for   19 
hours,  per  day.     The  all-day  efficiency  is 

watt  hours  output 


watt  hours  input 


per  day. 


5        ..  + r  +  24          + 

with  non-inductive  load,  Is,  being  the  full-load  secondary 
current. 

XII.    To  determine  the  regulation  in  per  cent. —  In  §  53 
was  shown  the  method  of  calculating  the  magnetic  leakage 


140  ALTERNATING-CURRENT    MACHINES. 

of  this  type  of  transformer.  Call  the  flux  linking  only  the 
primary  coils  <£,  (this  is  twice  that  which  links  the  coil  of 
one  limb  of  the  transformer).  Call  that  which  links  only 
the  secondary  coils  $„.  There  is  practically  no  voltage 
drop  at  no  load,  so  -Et=  rEp.  At  full  load  there  is  a  drop 
in  the  primary  and  in  the  secondary,  due  (a)  to  IR  drop, 
(fr)  to  self-induction  caused  by  leakage  flux.  Knowing 
this  leakage  flux,  by  the  formula  of  paragraph  L,  this  sec- 
tion, calculate  the  voltage  drop  in  primary  and  in  secondary 
coils,  thus, 


and  £sd  =  lo-»  \J 

The  regulation,  expressed  in  per  cent,  is 


Regulation  =  E'  ~  ^+  '•**  +  £~  .00, 

&, 

where  fp=Tftt  and  is  the  full-load  current.      Regulation  as 
stated  refers  to  a  non-inductive  load. 


MOTORS.  141 


CHAPTER   VII. 

MOTORS. 

60.  Rotating  Field.  —  Suppose  an  iron  frame,  as  in  Fig. 
Hi,  to  be  provided  with  inwardly  projecting  poles,  and  that 
these  be  divided  into  three  groups,  arranged  as  in  the  dia- 
gram, poles  of  the  same  group 
being  marked  by  the  same 
letter.  If  the  poles  of  each 
group  be  alternately  wound 
in  opposite  directions,  and  be 
connected  to  a  single  source 
of  E.M.F.,  then  the  resulting  current 
would  magnetize  the  interior  faces  al- 
ternately north  and  south.  If  the  im- 
pressed E.M.F.  were  alternating,  then 
the  polarity  of  each  pole  would  change  Fis-  x«- 

with  each  half  cycle.  If  the  three  groups  of  windings 
be  connected  respectively  with  the  three  terminals  of  a 
three-phase  supply  circuit,  any  three  successive  poles  will 
assume  successively  a  maximum  polarity  of  the  same 
sigh,  the  interval  required  to  pass  from  one  pole  to  its 
neighbor  being  one-third  of  the  duration  of  a  half  cycle. 
The  maximum  intensity  of  either  polarity  is  therefore 
passed  from  one  pole  to  the  next,  and  the  result  is  a  rotat- 
ing field.  If  the  frequency  of  the  supply  E.M.F.  be/,  and 
if  there  be  /  pairs  of  poles  per  phase,  then  the  field  will 


142  ALTERNATING-CURRENT   MACHINES. 

make  one  complete  revolution  in  -  seconds.     It  will  there- 

/       V 
fore  make  -  =  ^-  complete  revolutions    per    second.     A 

rotating  field  can  be  obtained  from  any  polyphase  supply- 
circuit  by  making  use  of  appropriate  windings. 

61.  The  Induction  Motor If  a  suitably  mounted  hollow 

conducting  cylinder  be  placed  inside  a  rotating  field,  it  will 
have  currents  induced  in  it,  due  to  the  relative  motion  be- 
tween it  and  the  field  whose  flux  cuts  the  surface  of  the 
cylinder.  The  currents  in  combination  with  the  flux  will 
react,  and  produce  a  rotation  of  the  cylinder.  As  the  cur- 
rent is  not  restrained  as  to  the  direction  of  its  path,  all  of 
the  force  exerted  between  it  and  the  field  will  not  be  in 
a  tangential  direction  so  as  to  be  useful  in  producing  rota- 
tion. This  difficulty  can  be  overcome  by  slotting  the 
cylinder  in  a  direction  parallel  with  the  axis  of  revolution. 
Nor  will  the  torque  exerted  be  as  great  as  it  would  be  if 
the  cylinder  were  mounted  upon  a  laminated  iron  core. 
Such  a  core  would  furnish  a  path  of  low  reluctance  for  the 
flux  between  poles  of  opposite  sign.  The  flux  for  a  given 
magnetomotive  force  would  thereby  be  greater,  and  the 
torque  would  be  increased. 

Induction  motors  operate  according  to  these  principles. 
The  stationary  part  of  an  induction  motor  is  called  the  stator, 
and  the  moving  part  is  called  the  rotor.  It  is  common 
practice  to  produce  the  rotating  field  by  impressing  E.M.F. 
upon  the  windings  of  the  stator.  There  are,  however, 
motors  whose  rotating  fields  are  produced  by  the  currents 
in  the  rotor  windings. 

Fig.  1 1 2  shows  the  stator  core  and  frame  of  a  Westing- 
house  induction  motor,  and  Fig.  1 1 3  shows  the  same  with 


MOTORS. 


143 


the  windings  in  place.  Each  projection  of  the  core  does 
not  necessarily  mean  a  pole  ;  for  it  is  customary  to  employ 
a  distributed  winding,  there  being  several  slots  per  pole 


Fig.  112.  Fig.  113. 

per  phase.      Fig.  1 14  shows  the  rotor.     The  inductors  are 

copper  bars  embedded  in  slots  in  the  laminated  steel  core. 

They  are  all  connected,  in  parallel,  to  copper  collars  or 

short-circuiting  rings,  one  at 
each  end  of  the  rotor.  They 
offer  but  a  very  small  resist- 
ance, and  the  currents  induced 
in  them  are  forced  to  flow  in 
a  direction  parallel  with  the 
axis.  The  reaction  against  the 
field  flux  is  therefore  in  a 
proper  direction  to  be  most 

efficient    in  producing  rotation.     A  rotor  or  armature  of 

this  type  is  called  a  squirrel  cage. 

62.   Principle  of  Operation  of  the  Induction  Motor.  —  If 
the  speed  of  rotation  of  the  field  be  V  R-  P.  M,  and  that  of 


Fig.  114. 


144          ALTERNATING-CURRENT    MACHINES. 

the  rotor  be  V  R.  P.  M.,  then  the  relative  speed  between  a 
given  inductor  on  the  rotor  and  the  rotating  field  will  be 
V—  V  R.  P.  M.  The  ratio  of  this  speed  to  that  of  the  field, 

V—  Vf 

viz., — —  =  s,  is  termed  the  slip,  and  is  generally  ex- 
pressed as  a  per  cent  of  the  synchronous  speed.  If  the 
flux  from  a  single  north  pole  of  the  stator  be  <£  maxwells, 
then  the  effect ive  E.M.F.  induced  in  a  single  rotor  inductor 

•t  r 

is    2.22  /3>  s —-  io~8,   where/  represents,  the  number  of 
60 

pairs  of  revolving  poles.  The  frequency  of  this  induced 
E.M.F.  is  different  from  that  of  the  E.M.F.  impressed  upon 
the  stator.  It  is  s  times  the  latter  frequency.  The  fre- 
quency would  be  zero  if  the  rotor  revolved  in  synchronism 
with  the  field,  and  would  be  that  of  the  field  current  if  the 
rotor  were  stationary.  As  the  slip  of  modern  machines  is 
but  a  few  per  cent  (2%  to  15%),  the  frequency  of  the 
E.M.F.  in  the  rotor  inductors,  under  operative  conditions, 
is  quite  low.  The  current  which  will  flow  in  a  given  in- 
ductor of  a  squirrel-cage  rotor  is  difficult  to  determine.  All 
the  inductors  have  E.M.F.' s  in  them,  which  at  any  instant 
are  of  different  values,  and  in  some  of  them  the  current 
may  flow  in  opposition  to  the  E.M.F.  It  can  be  seen,  how- 
ever, that  the  rotor  impedance  is  very  small.  As  the  im- 
pedance is  dependent  upon  the  frequency,  it  will  be  larger 
when  the  rotor  is  at  rest  than  when  revolving.  It  will  re- 
duce to  the  simple  resistance  when  the  rotor  is  revolving  in 
synchronism.  Suppose  a  rotor  to  be  running  light  without 
load.  It  will  revolve  but  slightly  slower  than  the  revolving 
field,  *so  that  just  enough  E.M.F.  is  generated  to  produce 
such  a  current  in  the  rotor  inductors  that  the  electrical 
power  is  equal  to  the  losses  due  to  friction,  windage,  and 


MOTORS.  145 

the  core  and  copper  losses  of  the  rotor.  If  now  a  me- 
chanical load  be  applied  to  the  pulley  of  the  rotor,  the 
speed  will  drop,  i.e.,  the  slip  will  increase.  The  E.M.F.  and 
current  in  the  rotor  will  increase  also,  and  the  rotor  will 
receive  additional  electrical  power,  equivalent  to  the  increase 
in  load.  The  induction  motor  operates  in  this  respect 
like  a  shunt  motor  on  a  constant  potential  direct-current 
'circuit.  If  the  strength  of  the  rotating  field,  which  cuts 


SYNCHRONISM 


Fig.  115. 

the  rotor  inductors,  were  maintained  constant,  the  slip, 
the  rotor  E.M.F.,  and  the  rotor  current  would  vary  directly 
as  the  mechanical  torque  exerted.  If  the  rotor  resistance 
were  increased,  the  same  torque  would  require  an  increase 
of  slip  to  produce  the  increased  E.M.F.  necessary  to  send 
the  same  current,  but  the  strict  proportionality  would  be 
maintained.  The  rotating  magnetism,  which  cuts  the  rotor 
inductors,  does  not,  however,  remain  constant  under  vary- 


146  ALTERNATING-CURRENT    MACHINES. 

ing  loads.  As  the  slip  increases,  more  and  more  of  the 
stator  flux  passes  between  the  stator  and  rotor  windings, 
without  linking  them.  This  increase  of  magnetic  leakage 
is  due  to  the  cross  magnetizing  action  of  the  increased 
rotor  currents.  The  decrease  of  linked  field  flux  not  only 
lessens  the  torque  for  the  same  rotor  current,  but  also 
makes  a  greater  slip  necessary  to  produce  the  same  cur- 
rent. The  relation  which  exists  between  torque  and  slip 
for  various  rotor  resistances  is  shown  in  Fig.  115,  where 
the  full  lines  represent  torque,  and  the  dotted  lines  current. 
An  inspection  of  the  curves  shows  that  the  maximum 
torque  which  a  motor  can  give  is  the  same  for  different 
rotor  resistances.  The  speed  of  the  rotor,  however,  when 
the  motor  is  exerting  this  maximum  torque,  is  different  for 
different  resistances.  This  fact  is  made  use  of  in  starting 
induction  motors  so  that  the  starting  current  may  not  be 
excessive.  Fig.  116  shows  a  General  Electric  Form  L 


Fig.  116. 


rotor.  The  winding  is  polar,  and  not  of  the  squirrel-cage 
type.  The  impedance  can  therefore  be  easily  calculated. 
The  terminals  of  the  windings  are  connected  to  a  resistance 
carried  on  the  rotor  spider.  When  the  rotor  reaches  a 


MOTORS. 


147 


proper  speed  the  resistance  may  be  cut  out  by  pushing  a 
knob  on  the  end  of  the  shaft,  as  shown  in  diagram.  This 
arrangement  permits  of  a  small  starting  current  under 
load  and  a  large  torque.  Squirrel-cage  motors  require 
several  times  full-load  current  to  start  under  load.  Fig. 


Fig.  117. 

117  shows  a  General  Electric  Co.  form  M  rotor.  The 
winding  is  the  same  as  in  the  Form  L,  except  that  its  ter- 
minals are  brought  out  to  three  slip-rings.  A  starting 
resistance  can  be  placed  away  from  the  motor  and  be  con- 
nected with  the  rotor  windings  by  means  of  brushes  rubbing 
upon  the  slip-rings. 

63.  The  Transformer  Method  of  Treatment. —  It  is  cus- 
tomary in  theoretical  discussions  to  consider  the  induction 
motor  as  a  transformer.  Evidently  when  the  rotor  is 
stationary  the  machine  is  nothing  but  a  transformer,  with 
a  magnetic  circuit  so  constructed  as  to  have  considerable 
magnetic  leakage.  When  the  rotor  is  moving,  the  machine 
still  acts  as  a  transformer  ;  but  the  ratio  of  transformation 
and  the  frequency  of  the  E.M.F.  in  the  rotor,  are  but  s 
times  what  they  were  with  a  stationary  rotor,  the  mechani- 
cal load  taking  the  place  of  the  electric  load  on  the  secon- 


148  ALTERNATING-CURRENT    MACHINES. 

clary  of  the  transformer.  Bearing  these  facts  in  mind,  the 
motor  may  be  treated  exactly  like  the  transformer.  Con- 
sider one  phase  of  a  polyphase  motor.  The  pressure  im- 
pressed upon  the  stator  is  greater  than  the  pressure  which 
is  operative  in  inducing  E.M.F.  in  the  rotor.  The  differ- 
ence is  due  to  the  resistance,  the  hysteresis,  the  eddy 
currents,  and  the  magnetic  leakage  of  the  stator.  The 
pressure  to  overcome  each  should  be  subtracted  from  the 
impressed  pressure  in  the  proper  phase  relation  to  get  the 
operative  pressure.  The  equivalent  inductance  of  the 
magnetic  leakage  can  be  calculated  for  different  currents, 
as  was  the  case  in  the  transformer.  The  voltage  induced 
in  the  rotor  is  sr  times  the  operative  pressure  of  the  stator 
where  T  is  the  ratio  of  transformation.  The  current  which 
it  produces  is  dependent  in  magnitude  and  phase  upon  the 
impedance  of  the  rotor  windings.  From  the  power  repre- 
sented by  this  current  at  the  rotor  pressure  must  be 
subtracted  the  power  lost  in  resistance,  eddy  currents,  hys- 
teresis, friction,  and  windage  of  the  rotor.  What  remains 
is  given  out  by  the  motor  as  useful  mechanical  power.  It 
should  not  be  forgotten  that  the  frequency  of  the  rotor 
currents  is  but  s  times  that  of  the  impressed  voltage. 

64.  Ratio  of  Transformation The  ratio  of  transforma- 
tion in  an  induction  motor  is  without  appreciable  effect 
upon  its  operation.  For  motors  of  the  same  capacity  it  is 
the  practice  of  the  General  Electric  Company  to  use  the 
same  squirrel-cage  rotor  for  different  voltages  and  different 
phases.  The  stator  windings  alone  are  altered.  Forms  L 
and  M  rotors  are  not  changed  for  change  of  voltage,  but 
must  of  course  be  altered  for  change  of  phase,  as  they  are 
polar  wound.  A  certain  4-pole,  3 -phase,  6ocycle,  no- 


MOTORS. 


149 


volt,  i -horse-power,  General  Electric  induction  motor  has 
36  slots  in  the  stator,  each  slot  containing  20  conductors  of 
size  No.  13.  The  rotor  contains  37  slots,  each  one  con- 
taining one  No.  2  wire.  The  slots  are  staggered  by  an 
amount  equal  to  the  distance  between  centers  of  two  con- 
secutive slots.  The  rotor  inductors  are  connected  to  short- 
circuit  disks,  one  on  each  end  of  the  rotor. 

65.  Behavior  of  Induction  Motors The  relations  be- 
tween speed,  torque,  power  factor,  efficiency,  and  current 
in  the  case  of  a  typical  induction  motor  operating  under 
normal  conditions  is  represented  in  Fig.  118. 

If  the.  voltage  impressed  upon  an  induction  motor  be 
increased,  there  will  result  a  proportional  increase  in  the 
flux  linked  with  the  rotor,  and  in  consequence  a  propor- 


10   20   30   40   50 


70   80   90   100  11.0  120  H.P. 


Fig.  118. 

tional  increase  in  the  rotor  current.  As  the  torque  de- 
pends upon  the  product  of  the  flux  and  the  rotor  ampere 
turns,  it  follows  that  the  torque  varies  as  the  square  of  the 
impressed  voltage.  The  capacity  of  a  motor  is  therefore 


150  ALTERNATING-CURRENT   MACHINES. 

changed  when  it  is  operated  on  circuits  of  different  volt- 
ages. 

Owing  to  the  low-power  factor  of  induction  motors, 
transformers  intended  to  supply  current  for  their  operation 
should  have  a  higher  rated  capacity  than  that  of  the  mo- 
tors. It  is  customary  to  have  the  kilowatt  capacity  of  the 
transformer  equal  to  the  horse-power  capacity  of  the  motor. 

The  low  power-factor  is  due  to  magnetic  leakage,  i.e., 
flux  linked  with  the  stator,  but  not  with  the  rotor  wind- 
ings. This  leakage  increases  with  increase  of  length  of 
air  gap.  It  is  hence  desirable  to  have  the  gap  as  small  as 
consistent  with  mechanical  clearance.  Concentricity  of 
rotor  and  stator  is  to  be  obtained  by  making  the  bearings 
in  the  form  of  end  plates  fastened  to  the  stator  frame. 
Some  makers  send  wedge  gap-gauges  with  their  machines 
so  that  a  customer  may  test  for  eccentricity  due  to  wear 
of  the  bearings.  A  small  air  gap,  besides  lowering  the 
leakage  and  raising  the  power  factor,  increases  the  effi- 
ciency and  capacity  of  the  motor. 

The  torque  exerted  on  a  constant  loaded  rotor  is  con- 
tinuous and  constant  in  the  case  of  a  polyphase  motor. 

The  Stanley  Company  raise  the  power  factor  of  their 
two-phase  motors  to  nearly  unity  by  using  condensers  to 
neutralize  the  lag  produced  by  leakage. 

The  direction  of  rotation  of  a  three-phase  motor  can  be 
changed  by  transposing  the  supply  connections  to  any  two 
terminals  of  the  motor.  In  the  case  of  a  two-phase,  four- 
wire  motor,  the  connections  to  either  one  of  the  phases, 
may  be  transposed. 

66.  Starting  of  Squirrel-Cage  Motors —  To  avoid  the 
excessive  rush  of  current  which  would  result  from  connec- 


MOTORS. 


tion  of  a  loaded  squirrel-cage  motor  to  a  supply  circuit,  use 
is  made  by  both  the  Westinghouse  Company  and  the 
General  Electric  Company  of  starting  compensators.  These 
are  autotransformers  which  are  connected  between  the 
supply  mains,  and  which,  through  taps,  furnish  to  the 
motor  circuits  currents  at  a  lower  voltage  than  that  of 
the  supply  mains.  After  the  rotor  has  attained  the  speed 
appropriate  to  the  higher  voltage,  the  motor  connections  are 
transferred  to  the  mains,  and  the  compensator  is  thrown 


nerator 

Running 

Side 

Oil  Switch 


Starting  Sid 


O-1  O-1  CM  O-1  CM  O-1 


Taps 


Compensator  Winding 

Fig.  11$. 

out  of  circuit.  The  connections  are  shown  in  Fig.  119, 
and  the  appearance  of  the  General  Electric  Company  com- 
pensator is  shown  in  Fig.  120.  The  change  of  connec- 
tions is  accomplished  by  moving  the  handle  shown  at  the 
right  of  the  figure.  While  the  compensator  is  supplied 
with  various  taps,  only  that  one  which  is  most  suitable  for 
the  work  is  used  when  once  installed.  The  Westinghouse 
compensator  is  shown  in  Fig.  121.  When  the  handle  is 
down  on  one  side,  the  autotransformers  are  in  circuit,  and 


152  ALTERNATING-CURRENT   MACHINES. 

the  motor  is  connected  with  the  low-voltage  taps.  Upon 
throwing  over  the  switch  the  transformers  are  cut  out,  and 
the  motor  is  connected  directly  with  the  mains. 


Where  special  step-down  transformers  are  used  for  indi- 
vidual motors,  or  where  several  motors  are  located  close  to 
and  operated  from  a  bank  of  transformers,  it  is  sometimes 
practical  to  bring  out  taps  from  the  secondary  winding,  and 
use  a  double-throw  motor  switch,  thereby  making  provision 
for  starting  the  motor  at  low  voltage,  while  avoiding  the 
necessity  for  a  starting  compensator. 

The  General  Electric  Company  make  small  squirrel- 
cage  motors,  with  centrifugal  friction  clutch  pulleys ;  so 
that  although  a  load  may  be  belted  to  the  motor,  it  is  not 
applied  to  the  rotor  until  the  latter  has  reached  a  certain 
speed.  The  starting  current  is  therefore  a  no-load  starting 
current. 


MOTORS. 


Fig.  121. 


67.  Phase  Splitters In  order  to  operate  polyphase  in- 
duction motors  upon  single-phase  circuits,  use  is  made  of 
inductances  in  series  with  one  motor  circuit  to  produce  a 


1.54 


ALTERNATING-CURRENT    MACHINES. 


Fig.  122. 

lagging  current,  or  of  condensers  to  produce  a  leading  cur- 
rent, or  of  both  —  one  in  each  of  two  legs.  The  General 
Electric  Company,  in  its  condenser  compensator,  for  use 
with  small  motors,  as  shown  in  Fig.  122,  employs  an 
autotransformer  and  condenser  connected, 
as  in  diagram  Fig.  123. 

The  autotransformer  is  used  to  step-up 
the  voltage,  which  is  impressed  upon  the 
condenser,  to  500  volts.  The  necessary 
size  of  the  condenser  is  thereby  reduced. 
The  equivalent  impedance  of  the  auto- 
transformer and  condenser,  as  connected, 
is  such  as  to  produce  a  leading  current  in 
the  one-phase  sufficient  to  give  a  satisfac- 
tory starting  torque,  and  it  brings  the  power 
factor  practically  up  to  unity  at  all  loads. 

68.  Single-Phase  Induction  Motors. —  A  two-phase  induc- 
tion motor  will  operate  fairly  well,  if,  after  attaining  full 


MOTORS. 


155 


speed,  one  of  the  two  phases  be  disconnected  from  the  sup- 
ply circuit.  It  will  not  start  from  rest  under  the  influence 
of  the  one-phase  excitation.  The  load  remaining  constant, 
the  one  phase  will  take  twice  its  original  current.  Simi- 
larly, a  three-phase  motor  will  operate  well  upon  one-phase 
excitation.  The  current  in  this  case  will  be  1.5  times 
what  it  previously  was.  A  motor  consisting -of  a  rotor- 


and  a  stator  wound  single  phase  will,  in  a  like  manner, 
operate  satisfactorily  when  once  started.  In  the  Wagner 
single-phase  induction  motor  (Fig.  124),  the  rotor  windings 
are  connected  to  a  commutator,  with  its  brushes  joined 
together  by  a  conductor  of  low  resistance.  The  stator 
is  supplied  with  single-phase  excitation.  The  rotor  is 
brought  up  to  speed  by  the  reaction  between  the  current 
which  is  induced  in  the  rotor  windings  and  the  stator  flux. 


156 


ALTERNATING-CURRENT   MACHINES. 


Upon  reaching  speed,  a  centrifugal  device,  shown  in  the 
figure,  causes  the  commutator  bars  to  be  short-circuited, 
and  the  brushes  are  simultaneously  lifted  from  the  commu- 


3  H.P.  SINGLE  PHASE 
INDUCTIONTMOTOR 

FROM  TESTS  MADE  AT 
HARVARD  UNIVERSITY, 

MAY  1900 

140  VOLTS 


50        60        70 
#FULL  LOAD 


Fig.  125. 

tator.  Tests  have  been  made  upon  this  type  of  motor  at 
various  universities,  including  Harvard,  University  of  Illi- 
nois, and  Purdue  University.  The  results  are  concordant, 
and  are  represented  in  the  curves  Fig.  125. 

69.  The  Monocyclic  System.  —  This  is  a  system  advocated 
by  the  General  Electric  Company  for  the  use  of  plants 
whose  load  is  chiefly  lights,  but  which  contains  some 
motors.  The  monocyclic  generator  is  a  modified  single- 
phase  alternator.  In  addition  to  its  regular  winding,  it 
has  a  so-called  teazer  winding,  made  of  wire  of  suitable 
cross  section  to  carry  the  motor  load,  and  with  enough 
turns  to  produce  a  voltage  one-fourth  that  of  the  regular 
winding,  and  lagging  90°  in  phase  behind  it.  One  end  of 
the  teazer  winding  is  connected  to  the  middle  of  the 


MOTORS.  157 

lar  winding,  and  the  other  end  is  connected  through  a  slip- 
ring  to  a  third  line  wire. 

A  three-terminal  induction  motor  is  used,  the  terminals 
being  connected  to  the  line  wires  either  directly  or  through 
transformers. 

70.  Frequency  Changers These  are  machines  which 

are  used  to  transform  alternating  currents  of  one  frequency 
into  those  of  another  frequency.     They  are  commonly  used 
to  transform  from  a  low  frequency  (say  from  25   or  40)  to 
a  higher  one.     They  depend  for  their  operation  upon  the 
variation  with  slip  of  the  frequency  of  the  rotor  E.M.F.'s 
of  an  induction  motor.     The  common  practice  for  raising 
the  frequency  is,  to  have  a  synchronous  motor  turn  the 
rotor  of  an  induction  motor  in  a  direction  opposite  to  the 
direction  of  rotation  of  the  latter's  field.     The  synchronous 
motor  and  the  stator  windings  of  the  induction  motor  are 
connected  to  the  low  frequency  supply  mains.     Slip-rings 
connected  to  the  rotor  windings  of  the  induction  motor 
supply  current  at  the  higher  frequency.     The  size  of  the 
synchronous    motor    necessary   to    drive    the    frequency 
changer  is  the  same  percentage  of  the  total  output  as  the 
rise  of  frequency  is  to  the  higher  frequency. 

71.  Speed  Regulation  of  Induction  Motors.  —  The  speed 
of  an  induction  motor  can  be  varied  by  altering  the  voltage 
impressed  upon  the  stator,  by  altering  the  resistance  of  the 
rotor  circuit,  or  by  commutating  the  stator  windings  so  as 
to  alter  the  multipolarity.     The  first  two  methods  depend 
for  their  operation  upon  the  fact  that,   inasmuch  as  the 
motor  torque  is  proportional  to  the  product  of  the  stator 
flux  and  the  rotor  current,  for  a  given  torque  the  product 


158  ALTERNATING-CURRENT    MACHINES. 

must  be  constant.  Lessening  the  voltage  impressed  upon 
the  stator  lessens  the  flux,  and  also  the  rotor  current,  if  the 
same  speed  be  maintained.  The  speed,  therefore,  drops 
until  enough  E.M.F.  is  developed  to  send  sufficient  current 
to  produce,  in  combination  with  the  reduced  flux,  the 
equivalent  torque.  Increasing  the  resistance  of  the  rotor 
circuit  decreases  the  rotor  current,  and  requires  a  drop  in 
speed  to  restore  its  value.  Both  of  these  methods  result 
in  inefficient  operation.  If  the  impressed  voltage  be  re- 
duced, the  capacity  of  the  motor  is  reduced.  In  fact,  the 
capacity  varies  as  the  square  of  the  impressed  voltage. 
Changes  in  the  multipolarity  of  the  stator  requires  compli- 
cated commutating  devices. 

72.  Synchronous  Motors.  —  Any  excited  single-phase 
or  polyphase  alternator,  if  brought  up  to  speed,  and  if  con- 
nected with  a  source  of  alternating  E.M.F.  of  the  same 
frequency  and  approximately  the  same  pressure,  will  ope- 
rate as  a  motor.  The  speed  of  the  rotor  in  revolutions 
per  second  will  be  the  quotient  of  the  frequency  by  the 
number  of  pairs  of  poles.  This  is  called  the  synchronous 
speed  ;  and  the  rotor,  when  it  has  this  speed,  is  said  to  be 
running  in  synchronism.  This  exact  speed  will  be  main- 
tained throughout  wide  ranges  of  load  upon  the  motor  up 
to  several  times  full-load  capacity. 

To  understand  the  action  of  the  synchronous  motor, 
suppose  it  to  be  supplied  with  current  from  a  single 
generator. 

Let   £1  =  E.M.F.  of  the  generator,  - 

E*  =  E.M.F.  of  the  motor  at  the  time  of  connec- 
tion with  the  generator,  - 
6  =  Phase  angle  between  Ev  and  E^ 


MOTORS.  159 

jR  =  Resistance  of  generator  armature,  plus  that 
of   the    connecting  wires  and    of   the 
motor  armature,  and 
o>Z  =  Reactance  of  the  above. 

The  resultant  E.M.F.,  E  which  is  operative  in  sending 
current  through  the  complete  circuit,  is  found  by  combin- 
ing El  and  E2  with  each 
other  at  a  phase  differ- 
ence 0,  as  in  Fig.  126. 

Representing  the 
angle  between  E^  and  E 
and  E9  and  E  by  a  and 
ft  respectively,  it  follows  that 

E  =  £l  cos  a  -f-  E%  cos  ft. 

This    resulting    E.M.F.    sends    through    the    circuit    a 
current  whose  value  is 

/=  ^  — 

and  it  lags  behind  E  by  an  angle  </>,  such  that  tan  <£  =  —  . 
The  power  Pl  which  the  generator  gives  to  the  circuit  is 

and  the  power  P.,  which  the  motor  gives  to  the  circuit  is 
P2  =  £2Scos  (ft  +  <£). 

Now,  if  in  either  of  the  above  expressions  for  power,  the 
cosine  has  any  other  value  than  unity,  then  the  power 
will  consist  of  energy  pulsations,  there  being  four  pulsa- 
tions per  cycle.  The  energy  is  alternately  given  to  and 
received  from  the  circuit  by  the  machine.  If  the  cosine 
be  positive,  the  amount  of  energy  in  one  pulsation,  which 


160  ALTERNATING-CURRENT   MACHINES. 

is  given  to  the  circuit,  will  exceed  the  amount  in  one 
of  the  received  pulsations.  The  machine  is  then  acting 
as  a  generator.  If  the  cosine  be  negative  the  opposite 
takes  place,  and  the  machine  operates  as  a  motor.  As  a 
and  ft  are  but  functions  of  E^  E2,  and  6,  and  as  these  latter 
are  the  quantities  to  be  considered  in  operation,  it  is  desir- 
able to  eliminate  the  former.  By  a  somewhat  complicated 
analytical  transformation  it  can  be  shown  that 


J\  = 


If  there  were  no  losses  due  to  resistance,  etc.,  Pl  would  be 
numerically  exactly  equal  to  Pz.  Neglecting  any  losses 
in  the  machines,  except  that  due  to  resistance,  the  alge- 
braic sum  of  Pl  and  P2  is  equal  to  RI*.  In  order  to 
determine  the  behavior  of  a  synchronous  motor  when  on 
a  given  circuit,  use  is  made  of  the  above  formula  for  power, 
and  each  case  must  be  considered  by  itself.  The  method 
of  procedure  is  shown  in  the  next  article. 

73.  Special  Case.  —  Suppose  a  single-phase  synchronous 
motor,  excited  so  as  to  generate  2100  volts,  to  be  con- 
nected to  a  generator  giving  2200  volts,  the  total  resis- 
tance of  the  circuit  being  2  ohms  and  the  reactance  i  ohm. 
Then  the  angle  <£  of  current  lag  behind  the  resultant 

E.M.F.  has  a  value  tan  <f>  =  —  -  =  0.5,  whence  </>  =  26°  34'. 

R 

A  preliminary  calculation,  using  the  formulas  of  the  pre- 
vious article,  shows  that  both  machines  act  as  generators 
for  values  of  0  between  o°  and  120°,  and  between  240°  and 
360°  approximately. 


MOTORS. 


161 


Calculations  of  P1  and  P2  for  various  values  of  0  between 
1 20°  and  240°  have  been  made,  and  are  embodied  in  the 
form  of  curves  in  Fig.  127.  From  an  inspection  of  these 


Fig.  137. 

curves,  and  a  consideration  of  the  equations  from  which 
the  curves  are  derived,  the  following  conclusions  may  be 
drawn  :  — 

(a)  The  motor  will  operate  as  such  for  values  of  0  be- 
tween   175°   and    238°.      The   difference    between    these 
angles  may  be  termed  the  operative  range. 

(b)  The  generator  would  operate  as  a  motor  for  values 
of  0  between   133°  and   174°,  providing  the   motor   were 
mechanically  driven  so  as  to  supply  the  current  and  power  ; 
i.e.,  what  was  previously  the  motor  must  now  operate  as 
a  generator. 

(c)  The  motor,  within  its  operative  range,  can  absorb 
any  amount  of    power  between  zero  and  a  certain  maxi- 
mum.    To  vary  the  amount  of  received  power,  the  motor 
has  to  but  slightly  shift  the  phase  of  its  E.M.F.  in  respect 
to  the  impressed  E.M.F. ,  and  then  to  resume  running  in 


162  ALTERNATING-CURRENT   MACHINES. 

synchronism.  The  sudden  shift  of  phase  under  change 
of  load  is  the  fundamental  means  of  power  adjustment  in 
the  synchronous  motor.  It  corresponds  to  change  of  slip 
in  the  induction  motor,  to  change  of  speed  in  the  shunt 
motor,  and  to  change  of  magnetomotive  force  in  the 
transformer. 

(d)  For  all  values  of  the  received  power,  except  the 
maximum,  there  are  two  values  of  phase  difference  0. 
At  one  of  these  phase  differences  more  current  is  required 
for  the  same  power  than  at  the  other.  The  value  of  the 
current  in  either  case  can  be  calculated  as  follows  :  — 

Since  /\  +  Pz  = 


The  values  of  /are  plotted  in  the  diagram.     The  efficiency 

p 
of  transmission  e  =  — *  is  also  different  for  the  two  values 

of  0  .     It  is  also  represented  by  a  curve. 

If  the  phase  alteration,  produced  by  an  added  mechan- 
ical load  on  the  motor,  results  in  an  increase  of  power 
received  by  the  motor,  the  running  is  said  to  be  stable.  If 
on  the  other  hand,  the  increase  of  load  produces  a  decrease 
of  absorbed  power,  the  running  is  unstable. 

(e)  If  for  any  reason  the  phase  difference  0,  between 
the  E.M.F.'s  of  the  motor  and  generator,  be  changed  to  a 
value  without  the  operative  range  for  the  motor,  the  motor 
will  cease  to  receive  as  much  energy  from  the  circuit  as  it 
gives  back,  and  it  will,  therefore,  fall  out  of  step.  Among 
the  causes  which  may  produce  this  result  are  sudden 
variations  in  the  frequency  of  the  generator,  variations  in 
the  angular  velocity  of  the  generator,  or  excessive  me- 


MOTORS.  163 

chanical  load  applied  to  the  motor.  In  slowing  down,  all 
possible  values  of  0  will  be  successively  assumed  ;  and  it 
may  happen  that  the  motor  armature  may  receive  suffi- 
cient energy  at  some  value  of  0  to  check  its  fall  in  speed, 
and  restore  it  to  synchronism,  or  it  may  come  to  a  stand- 
still. 

(/)  Under  varying  loads  the  inertia  of  the  motor 
armature  plays  an  important  part.  The  shifting  from  one 
value  of  0  to  another,  which  corresponds  to  a  new  mechan- 
ical load,  does  not  take  place  instantly.  The  new  value  is 
overreached,  and  there  is  an  oscillation  on  both  sides  of 
its  mean  value.  This  oscillation  about  the  synchronous 
speed  is  termed  hunting.  If  the  armature  required  no 
energy  to  accelerate  or  retard  it,  this  would  not  take 
place. 

(g)  The  maximum  negative  value  of  P2  —  that  is,  the 
maximum  load  that  the  motor  can  carry  —  is  evidently  when 
cos  (  0  —  <f>  )  =  —  i  or  when  9  —  <j>  =  1  80°.  The  formula 
for  the  power  absorbed  by  the  motor  then  reduces  to 

ZL2  cos  <f>  —  E,  E» 


(h)  The  operative  range  of  the  motor  can  be  determined 
by  making  P,2  equal  to  zero.  By  transformation  the  for- 
mula then  becomes 

E?  cos  <f> 

cos  (6  -  <£)  =  -       '         y  - 


Two  values  of  (0  —  <£)  result,  one  on  each  side  of  1  80°. 
In  the  case  under  consideration  cos  (6  —  <£  )  =  —  .851  and 
0-  <fr=  2  n°  40'  or  148°  20'.  Since  c6  «  26°  34',$=  238° 
14'  or  174°  54'. 


1 64 


ALTERNATING-CURRENT    MACHINES. 


74.  The  Motor  E.M.F.  —  To  determine  what  value  of 
Ez  will  give  the  maximum  value  of  power  to  be  absorbed 
by  a  motor,  consider  E2  as  a  variable  in  the  equation  given 
in  (g)  above. 

Differentiating 


cos  t>  — 


and  setting  this  equal  to  zero  and  solving, 
JE«  = —  =  1210  volts. 

2   COS  <£ 

At  this  voltage  the  maximum  possible  intake  of  the  motor 
is  6 1 1  K.  w.  If  the  voltage  of  the  motor  be  above  this  or 
below  it,  its  maximum  intake  will  be  smaller. 

Remembering  that  the  current  lags  behind  the  resultant 
pressure  of  the  generator  and  motor  pressures  by  an  angle 
cf>,  which  is  solely  dependent  upon  to, 
L,  and  R,  it  will  be  easily  seen,  from 
an  inspection  of  Figs.  128,  129,  and 
1 30,  that  the  current  may  be  made 
to  lag  behind,  lead,  or  be  in  phase 
with  E^  by  simply  altering  the  value 
of  E2.  This  may  be  done  by  vary- 
ing the  motor's  field  excitation.  A 
proper  excitation  can  produce  a  unit 
power  factor  in  the  transmitting 
line.  The  over-excited  synchronous 
motor,  therefore,  acts  like  a  con- 
denser in  producing  a  leading  cur- 
rent, and  can  be  made  to  neutralize  the  effect  of  induct- 
ance. The  current  which  is  consumed  by  the  motor  for  a 
given  load  accordingly  varies  with  the  excitation.  The 


CURRENT  LAGGING  E, 

Fig.  128. 


CURRENT  LEADING  E, 

Fig.  129. 


CURRENT  IN  PHASE  WITH  E( 

Fig.  130. 


MOTORS. 


I65 


relations  between  motor  voltage  and  absorbed  current  for 
various  loads  are  shown  in  Fig.  131. 

Synchronous  motors  are  sometimes  used  for  the  purpose 
of  regulating  the    phase   relations  of   transmission   lines. 


MOTOR  VOLTAGE 


Fig.  131. 

The  excitation  is  varied  to  suit  the  conditions,  and  the 
motor  is  run  without  load.  Under  such  circumstances  the 
machines  are  termed  synchronous  compensators. 

The  capacity  of  a  synchronous  motor  is  limited  by  its 
heating.  If  it  is  made  to  take  a  leading  current  in  order 
to  adjust  the  phase  of  a  line  current,  it  cannot  carry  its 
full  motor  load  in  addition  without  heating. 

75.  Polyphase   Synchronous   Motor.  —  The    discussion 
which    has    just    been    given  applies    to   the    single-phase 
motor.     The  facts  brought  out  are  equally  applicable  to  the 
polyphase  motor.     In  the  latter  case  each  leg  or  phase  is 
to  be  considered  as  a  single-phase  circuit.    The  total  power 
is  that  of  each  phase  multiplied  by  the  number  of  phases. 

76.  Starting  Synchronous  Motors. — These  motors  do  not 
have  sufficient  torque  at  starting  to  satisfactorily  come  up 


1 66 


ALTERNATING-CURRENT    MACHINES. 


to  speed  under  load.  They  are,  therefore,  preferably 
brought  up  to  synchronous  speed  by  some  auxiliary  source 
of  power.  In  the  case  of  polyphase  systems  an  induction 
motor  is  very  satisfactory.  Its  capacity  need  be  but  TL  that 
of  the  large  motor.  Fig.  132  shows  a  750  K.  w.  quarter- 
phase  General  Electric  motor  with  a  small  induction  motor 


Fig.  132. 

geared  to  the  shaft  for  this  purpose.  This  motor  may  be 
mechanically  disconnected  after  synchronism  is  reached. 
Before  connection  of  the  synchronous  motor. to  the  mains 
it  is  necessary  that  the  motor  should  not  only  be  in  syn- 
chronism, but  should  have  its  electromotive  force  at  a 
difference  of  phase  of  about  180°  with  the  impressed 
pressure.  To  determine  both  these  points  a  simple  device, 


iMOTORS.  167 

known  as  a  synchronizer)  is  employed.  It  consists  of 
an  incandescent  lamp  connected  in  series  with  the  sec- 
ondaries of  two  transformers,  whose  primaries  are  con- 
nected respectively  with  the  line  and  with  the  motor 
terminals.  The  brightness  with  which  the  lamp  glows 
is  a  measure  of  the  phase  difference  between  the  two 
E.M.F.'s.  It  is  customary  to  so  connect  the  transformers 
that  when  the  motor  E.M.F.  is  at  180°  with  the  line 
pressure,  the  lamp  will  have  its  greatest  brilliancy.  As 
the  motor  is  coming  up  to  speed,  the  lamp  will  be 
alternately  bright  and  dark.  The  alternations  will  grow 
slower  as  synchronism  is  approached,  and  will  finally  be 
so  slow  as  to  permit  the  closing  of  the  main  switch  at  the 
proper  instant. 

Synchronous  motors  may  be  brought  up  to  speed  with- 
out any  auxiliary  source  of  power.  The  field  circuits  are 
left  open  ;  and  the  armature  is  connected  either  to  the  full 
pressure  of  the  supplv,  or  to  this  pressure  reduced  by 
means  of  a  starting  compensator,  such  as  was  described 
in  §  66.  The  magnetizing  effect  of  the  armature  ampere 
turns  sets  up  a  flux  in  the  poles  sufficient  to  supply  a  small 
starting  torque.  When,  after  running  a  sufficient  time  as  an 
induction  motor,  synchronism  is  nearly  attained,  the  fields 
may  be  excited  and  the  motor  will  come  into  step.  The 
load  is  afterwards  applied  to  the  motor  through  friction 
clutches  or  other  devices.  There  is  great  danger  of  per- 
forating the  insulation  of  the  field  coils  when  starting  in 
this  manner.  This  is  because  of  the  high  voltage  produced 
in  them  by  the  varying  flux.  In  such  cases  each  field 
spool  is  customarily  open-circuited  on  starting.  Switches 
which  are  designed  to  accomplish  this  purpose  are  called 
break-up  switches. 


168  ALTERNATING-CURRENT    MACHINES. 

77.  Parallel  Running  of  Alternators Any  two  alter- 
nators adjusted  to  have  the  same  E.M.F.,  and  the  same 
frequency,  may  be  synchronized  and  run  in  parallel.  Ma- 
chines of  low  armature  reaction  have  large  synchronizing 
power,  but  may  give  rise  to  heavy  cross  currents,  if  thrown 
out  of  step  by  accident.  The  contrary  is  true  of  machines 
having  large  armature  reaction.  Cross  currents  due  to 
differences  of  wave-form  are  also  reduced  by  large  arma- 
ture reaction.  The  electrical  load  is  distributed  between 
the  two  machines  according  to  the  power  which  is  being 
furnished  by  the  prime  movers.  This  is  accomplished,  as 
in  the  case  of  the  synchronous  motor,  by  a  slight  shift  of 
phase  between  the  E.M.F.'s  of  the  two  machines.  The 
difficulties  which  have  been  experienced  in  the  parallel 
running  of  alternators  have  almost  invariably  been  due  to 
bad  regulation  of  the  speed  of  the  prime  mover.  Trouble 
may  arise  from  the  electrical  side,  if  the  alternators  are 
designed  with  a  large  number  of  poles.  Composite  wound 
alternators  should  have  their  series  compounding  coils  con- 
nected to  equalizing  bus  bars,  the  same  as  compound  wound 
direct-current  generators. 


CONVERTERS. 


169 


CHAPTER   VIII. 

CONVERTERS. 

78.  The  Converter.  —  The  converter  is  a  machine  hav- 
ing one  field,  and  one  armature,  the  latter  being  supplied 
with  both  a  direct-current  commutator  and  alternating- 
current  slip-rings.  When  brushes,  which  rub  upon  the 
slip-rings,  are  connected  with  a  source  of  alternating 
current  of  proper  voltage,  the  armature  will  rotate  syn- 
chronously, acting  the 
same  as  the  armature  of  a 
synchronous  motor.  While 
so  revolving,  direct  current 
can  be  taken  from  brushes 
rubbing  upon  the  commu- 
tator. The  intake  of  cur- 
rent from  the  alternating- 
current  mains  is  sufficient 
to  supply  the  direct -current 
circuit,  and  to  overcome 
the  losses  due  to  resistance, 
friction,  windage,  hyster- 
esis, and  eddy  currents.  The  windings  of  a  converter 
armature  are  closed,  and  simply  those  of  a  direct-current 
dynamo  armature  with  properly  located  taps  leading  to  the 
slip-rings.  Each  ring  must  be  connected  to  the  armature 
winding  by  as  many  taps  as  there  are  pairs  of  poles  in 
the  field.  These  taps  are  equidistant  from  each  other. 


Fig.  i33- 


ALTERNATING-CURRENT    MACHINES. 


There  may  be  any  number  of  rings  greater  than  one. 
A  converter  having  ;/  rings  is  called  an  «-ring  converter. 

The  taps  to  successive  rings  are  -th  of  the  distance  be- 
tween the  centers  of  two  successive  north  poles  from  each 
other.  Fig.  133  shows  the  points  of  tapping  for  a  3-ring 
multipolar  converter. 

A  converter  may  also  be   supplied  with  direct  current 


Fig.  134. 

through  its  commutator,  while  alternating  current  is  taken 
from  the  slip-rings.  Under  these  circumstances  the 
machine  is  termed  an  inverted  converter.  Converters  are 
much  used  in  lighting  and  in  power  plants,  sometimes 
receiving  alternating  current,  and  at  other  times  direct 
current.  In  large  city  distributing  systems  they  are  often 
used  in  connection  with  storage  batteries  to  charge  them 


CONVERTERS.  1 71 

from  alternating-current  mains  during  periods  of  light 
load,  and  to  give  back  the  energy  during  the  heavy  load. 
They  are  also  used  in  transforming  alternating  into  direct 
currents  for  electrolytic  purposes.  A  three-phase  machine 
for  this  purpose  is  shown  in  Fig.  1 34. 

A  converter  is  sometimes  called  a  rotary  converter  or 
simply  a  rotary. 

79.  E.M.F.  Relations.  —  In  order  to  determine  the  re- 
lations which  exist  between  the  pressures  available  at  the 
various  brushes  of  a  converter, 

Let  Ed  =  the  voltage  between  successive  direct-current 

brushes. 
En  =  the  effective  voltage  between  successive  rings 

of  an  //-ring  converter. 

a  =  the  maximum  E.M.F.  in  volts  generated  in  a 
single  armature  inductor.  This  .will  Sexist 
when  the  conductor  is  under  the  center  of  a 
pole. 

b  =  the  number  of  armature  inductors  in  a  unit 
electrical  angle  of  the  periphery.  The 
electrical  angle  subtended  by  the  centers  of 
two  successive  poles  of  the  same  polarity 
is  considered  as  2?r 

The  E.M.F.  generated  in  a  conductor  may  be  considered 
as  varying  as  the  cosine  of  the  angle  of  its  position  relative 
to  a  point  directly  under  the  center  of  any  north  pole,  the 
angles  being  measured  in  electrical  degrees.  At  an  angle 
/?,  Fig.  135,  the  E.M.F.  generated  in  a  single  inductor  G 
is  a  cos  (3  volts.  In  an  element  d$  of  the  periphery  of 
the  armature  there  are  bd$  inductors,  each  with  this 
E.M.F.  If  connected  in  series  they  will  yield  an  E.M.F. 


172 


ALTERNATING-CURRENT   MACHINES. 


of  ab  cos  ft  d$  volts.     The  value  of  ab  can  be  determined 
if  an  expression  for  the  E.M.F.  between  two  successive 

direct-current  brushes  be 
determined  by  integration, 
and  be  set  equal  to  this 
value  Ed  as  follows  : 


Fig.  135- 

2?r 

successive  rings  is 

n 


r+l 

••   I        ab  cos  fid(3  =  2  ab. 


In  an  //-ring  converter,  the 
electrical  angular  distance 
between  the  taps  for  two 

The  maximum  E.M.F.  will  be 


generated  in  the  coils  between  the  two  taps  for  the  succes- 
sive rings,  when  the  taps  are  at  an  equal  angular  distance 
from  the  center  of  a  pole,  one  on  each  side  of  it,  as  shown 
in  the  figure.  This  maximum  E.M.F.  is 


ab  cos  3dB  =  2  ab  sin  - 
^  ^  n 


=  Ed  sin  - 

n 


The  effective   voltage  between  the  successive  rings  is 
therefore 


By  substituting  numerical  values  in  this  formula,  it  is 
found  that  the  coefficient  by  which  the  voltage  between 


CONVERTERS. 


1/3 


the  direct-current  brushes  must  be  multiplied  in  order  to 
get  the  effective  voltage  between  successive  rings  is  for 

2  rings  .     .     .     .     ....     .     ."  .  0.707 

3  rings  ...-'.     .     .' 0.612 

4  rings 0.500 

6  rings  .    j     .     .     ......  0.354 


In  practice  there  is  a  slight  variation  from  these  co-effi- 
cients due  to  the  fact  that  the  air-gap  flux  is  not  sinusoid- 
ally  distributed. 

80.  Current  Relations. —  In  the  following  discussion  it  is 
assumed  that  a  converter  has  its  field  excited  so  as  to 
cause  the  alternating  currents  in  the  armature  inductors  to 
lag  1 80°  behind  the  alternating  E.M.F.  generated  in  them. 

The  armature  coils  carry  currents  which  vary  cyclically 
with  the  same  frequency  as  that  of  the  alternating-current 
supply.  They  differ 
widely  in  wave-form  from 
sine  curves.  This  is  be- 
cause they  consist  of  two 
currents  superposed  upon 
each  other.  Consider  a 
coil  B,  Fig.  136.  It  car- 
ries a  direct  current  whose 

value  —  is  half  that  car- 
2 

ried  by  one  direct-current 

brush,  and  it  reverses  its 

direction  every  time  that  Fig-  I36' 

the    coil    passes    under   a   brush.      The   coil,   as  well    as 

all  others  between  two  taps  for  successive  slip-rings,  also 

carries  an  alternating  current.     This  current  has  its  zero 


174 


ALTERNATING-CURRENT    MACHINES. 


value  when  the  point  A,  which  is  midway  between  the 
successive  taps,  passes  under  the  brush.  The  coil  being 
\l/  electrical  degrees  ahead  of  the  point  A,  the  alternating 

current  will  pass  through  zero  —  of  a  cycle    later   than 

2  7T 

the  direct  current.  The  time  relations  of  the  two  currents 
are  shown  in  Fig.  137. 

To   determine   the   maximum  value   of   the    alternating 
current  consider  that,  after  subtracting  the  machine  losses, 


Fig.  137. 


the  alternating-current  power  intake  is  equal  to  the  direct- 
current  power  output.  Neglecting  these  losses  for  the 
present,  if  En  represents  the  pressure  and  /„  the  effective 
alternating  current  in  the  armature  coils  between  the  suc- 
cessive slip-rings,  then  for  the  parts  of  the  armature  wind- 
ings covered  by  each  pair  of  poles 

Edld  =  nEJn 

Ed    .    TT 
=  n  —  -  sin  -  In  . 

V^       « 
Therefore,  the  maximum  value  of  the  alternating  current  is 


7T 

n  sin  - 
n 


The  time  variation  of  current   in  the  particular  coil  B  is 
obtained  by  taking  the  algebraic  sum  of  the  ordinates  of 


CONVERTERS. 


175 


the  two  curves.  This  yields  the  curve  shown  in  Fig.  138. 
Ivach  inductor  has  its  own  wave-shape  of  current,  depend- 
ing upon  its  angular  distance  ^  from  the  point  A. 


\ 


Fig.  138. 


81.  Heating  of  the  Armature  Coils The  heating  ef- 
fect in  an  armature  coil  due  to  a  current  of  such  peculiar 
wave-shape  as  that  shown  in  Fig.  138  can  be  determined 
either  graphically  or  analytically.  The  graphic  determina- 
tion requires  that  a  new  curve  be  plotted,  whose  ordinates 
shall  be  equal  to  the  squares  of  the  corresponding  current 
values.  The  area  contained  between  this  new  curve  and 
the  time  axis  is  then  determined  by  means  of  a  planimeter. 
The  area  of  one  lobe  is  proportional  to  the  heating  value 
of  the  current.  This  value  may  be  determined  for  each 
of  the  coils  between  two  successive  taps.  An  average  of 
these  values  will  give  the  average  heating  effect  of  the 
currents  in  all  the  armature  coils.  The  heating  is  different 
in  the  different  coils.  It  is  a  maximum  for  coils  at  the 
points  of  tap  to  the  slip-rings  and  is  a  minimum  for  coils 
midway  between  the  taps. 

The  analytical  determination  is  made  as  follows  :  For 
a  coil  which  is  ^  electrical  degrees  from  a  point  midway 


176          ALTERNATING-CURRENT   MACHINES. 

between  successive  slip-ring  taps,  at  the  time  t  seconds 
after  passing  a  direct-current  positive  brush,  the  instanta- 
neous value  of  the  current  is 

T,  _I*\  4  sin  (2  icft  —  \f)  _      1 

-/        —  I  I      J     " 

2    -{  .       7T 

n  sm  - 
I  n  J 

The  effective  value  of  the  current  in  this  coil  is,  therefore, 


1 6  cos  ^ 

_j— 


7T  .     _  7T 

sm  —      ^r  smj  - 
n  n 


where  Q  represents  for  simplicity  the  value  of  the  radical. 
The  heating,  due  to  the  current  in  this  coil,  is  propor- 

f2@2  ^ 

tional  to  -  LH. ,  and  the  average  heating  over  the  whole 

4 
armature  is  proportional  to 


Inasmuch  as  the  heating  of  this  armature  when  run  as  a 

simple  direct-current  generator  is  proportional  to  — ,  it 

4 

can,  with  the  same  heating,  when  operating  as  a  conver- 
ter, put  out  *  as  much  direct  current. 


CONVERTERS.  177 

82.  Capacity  of  a  Converter.  —  By   inserting   numerical 
values  in  the  above  equation  it  is  found  that  a  machine  has 
different  capacities,  based  upon  the  same  temperature  rise, 
according  to  the  number  of  slip-rings,  as  shown  in  the  fol- 
lowing table.     The  armature  is  supposed  to  have  a  closed- 
coil  winding  :  — 

CONVERTER  CAPACITIES. 
USED  AS  A  KILOWATT  CAPACITY 

Direct-current  generator i  oo 

Single-phase  converter 85 

Three-phase  converter 134 

Four-phase  converter 164 

Six-phase  converter 196 

Twelve-phase  converter 227 

The  overload  capacity  of  a  converter  is  limited  by  com- 
mutator performance  and  not  by  heating.  As  there  is  but 
small  armature  reaction,  the  limit  is  much  higher  than  is 
the  case  with  a  direct-current  generator. 

83.  Starting  a  Converter.  —  Converters  may  be  started 
and  be  brought  up  to  synchronism  by  the  same  methods 
which  are  employed  in  the  case  of  synchronous  motors. 
It  is  preferable,  however,  that  they  be  started  from  the 
direct-current  side  by  the  use  of  storage  batteries  or  other 
sources  of  direct  current.     They  may  be  brought  to  a  little 
above  synchronous  speed  by  means  of  a  starting  resistance 
as  in  the  case  of  a  direct-current  shunt  motor,  and  then, 
after  disconnecting  and  after  opening  the  field  circuit,  the 
connections   with  the    alternating-current    mains  may  be 
made.     This  will  bring  it  into  step. 

84.  Armature  Reaction.  —  The  converter  armature  cur- 
rents give  rise  to  reactions  which  consist  of  direct-current 


1 78 


ALTERNATING-CURRENT   MACHINES. 


generator  armature  reactions  superposed  upon  synchronous 
motor  armature  reactions.  It  proves  best  in  practice  to 
set  the  direct-current  brushes  so  as  to  commutate  the  cur- 
rent in  coils  when  they  are  midway  between  two  succes- 


Fig.  139. 

sive  poles.  The  direct-current  armature  reaction,  then,  con- 
sists in  a  cross-magnetization  which  tends  to  twist  the  field 
flux  in  the  direction  of  rotation.  When  the  alternating 
currents  are  in  phase  with  the  impressed  E.M.F.  they  also 
exert  a  cross-magnetizing  effect  which  tends  to  twist  the 


CONVERTERS.  179 

field  flux  in  the  opposite  direction.  The  result  of  this  neu- 
tralization is  a  fairly  constant  distribution  of  flux  at  all 
loads.  Within  limits  even  an  unbalanced  polyphase  con- 
verter operates  satisfactorily.  There  is  no  change  of  field 
excitation  necessary  with  changes  of  load. 

The  converter  is  subject  to  hunting  the  same  as  the 
synchronous  motor.  As  its  speed  oscillates  above  and 
below  synchronism,  the  phase  of  the  armature  current,  in 
reference  to  the  impressed  E.M.F.,  changes.  This  results 
in  a  distortion  of  the  field  flux,  of  varying  magnitude. 
This  hunting  is  much  reduced  by  placing  heavy  copper 
circuits  near  the  pole  horns  so  as  to  be  cut  by  the  oscillat- 
ing flux  from  the  two  horns  of  the  pole.  The  shifting  of 
flux  induces  heavy  currents  in  these  circuits  which  oppose 
the  shifting.  Fig.  139  shows  copper  bridges  placed  be- 
tween the  poles  of  a  converter  for  this  purpose. 

When  running  as  an  inverted  converter  from  a  direct- 
current  circuit,  anything  which  tends  to  cause  a  lag  of  the 
alternating  current  behind  its  E.M.F.  is  to  be  avoided. 
The  demagnetization  of  the  field  by  the  lagging  current 
causes  the  armature  to  race  the  same  as  in  the  case  of  an 
unloaded  shunt  motor  with  weakened  fields.  Converters 
have  been  raced  to  destruction  because  of  the  enormous 
lagging  currents  due  to  a  short  circuit  on  the  alternating- 
current  system. 

85.   Regulation  of  Converters The  field  current  of  a 

converter  is  generally  taken  from  the  direct-current 
brushes.  By  varying  this  current  the  power  factor  of  the 
alternating-current  system  may  be  changed.  This  may 
vary,  through  a  limited  range,  the  voltage  impressed 
between  the  slip-rings.  As  the  direct-current  voltage 


i8o 


ALTERNATING-CURRENT   MACHINES. 


Step-down 
Transformer. 


Regulator. 


bears  to  the  latter  a  constant  ratio  it  may  also  be  varied. 
This  is,  however,  an  uneconomical  method  of  regulation. 
Converters  are  usually  fed  through  step-down  transform- 
ers.    In    such    cases 
there    are    two    com- 
mon methods  of  regu- 
lation, which  vary  the 
voltage     supplied     to 
Fig-  I4°-  the    converter's    slip- 

rings.      The   first   is  the  method    of    Stillwell,   which   is 
shown  in  the  "diagram,  Fig.  140. 

The  regulator  consists  of  a  transformer  with  a  sectional 


Fig.  141. 


CONVERTERS.  l8l 

secondary.  Its  ratio  of  transformation  can  be  altered  by 
moving  a  contact -arm  over  blocks  connected  with  the 
various  sections,  as  shown  in  the  diagram.  The  primary 
of  the  regulator  is  connected  with  the  secondary  terminals 
of  the  step-down  transformer.  The  sections  of  the  second- 
ary, which  are  in  use,  are  connected  in  series  with  the  step- 
down  secondary  and  the  converter  windings. 

The  second  method  of  regulation  is  that  employed  by 
the  General  Electric  Co.  The  ratio  of  transformation  of 
a  regulating  transformer,  which  is  connected  in  circuit  in 
the  same  manner  as  the  Stillwell  regulator,  is  altered  by 
shifting  the  axes  of  the  primary  and  secondary  coils  in 
respect  to  each  other.  Fig.  141  shows  such  a  transformer, 
the  shifting  being  accomplished  by  means  of  a  small, 
direct-current  motor  mounted  upon  the  regulator.  The 
primary  windings  are  placed  in  slots  on  the  interior  of  a 
laminated  iron  frame,  which  has  the  appearance  of  the 
stator  of  an  induction  motor.  The  secondary  windings  are 
placed  in  what  corresponds  to  the  slots  of  the  rotor  core. 
The  winding  is  polar ;  and  if  the  secondary  core  be  rotated 
by  an  angle  corresponding  to  the  distance  between  two 
successive  poles,  the  action  of  the  regulator  will  change 
from  that  of  booster  to  that  of  crusher. 


182  ALTERNATING-CURRENT   MACHINES. 


CHAPTER   IX. 

POWER  TRANSMISSION. 

86.  Superiority  of  Alternating  Currents.  —  The  two 
great  sources  of  energy  for  use  in  manufacturing  establish- 
ments and  in  land  transportation  systems  are  the  coal 
mines  and  the  water  powers.  While  coal  can  be  trans- 
ported to  the  point  of  utilization  of  the  energy,  the 
energy  of  the  waterfall  cannot  be  commercially  transmitted 
to  a  long  distance  without  the  use  of  electricity.  In 
many  cases  it  is  uncertain  whether  it  is  not  cheaper  to 
transmit  the  energy  of  the  coal  in  the  form  of  electrical 
energy  than  to  transport  the  coal  itself.  There  is  gen- 
erally greater  convenience  and  greater  flexibility  in  the 
application  and  utilization  of  the  transmitted  electrical 
energy. 

The  electrical  transmission  can  be  accomplished  by 
means  of  direct  currents  or  by  means  of  alternating  cur- 
rents. For  transmission  over  anything  but  quite  short 
distances  the  alternating  current  is  preferable  to  the  direct 
current.  Even  for  short  distances,  when  these  pass 
through  densely  populated  districts,  the  alternating  cur- 
rent is  adopted  for  pure  transmission  purposes. 

The  direct  current  has  its  points  of  superiority.  Its 
use  is  not  attended  by  inductive  disturbances  with  the  ac- 
companying drop  and  sometimes  low  power  factor  ;  it  is 
attended  by  no  appreciable  capacity  effects ;  it  is  not 


POWER   TRANSMISSION.  183 

subject  to  electric  surgings,  which  sometimes  cause  insu- 
lation perforations,  short  circuits,  and  arcing.  It  permits 
the  use  of  direct-current  motors  with  their  Very  satisfac- 
tory operation  as  to  efficiency,  small  starting  current, 
overload  capacity,  and  speed  control.  Its  use  on  trans- 
mission lines  of  over  a  few  miles'  length  is  prohibited  by 
the  cost  of  the  line' which  it  necessitates.  As  will  be  seen 
later,  long  distance  electrical  power  transmission,  to  be 
economical  or  even  commercially  possible,  must  be  ef- 
fected by  high  voltages.  Direct-current  sparkless  com- 
mutation is  limited  to  1000  volts.  This  limit  is  dependent 
upon  the  economical  and  mechanical  limits  of  armature 
peripheral  velocity,  current  density,  gap-flux  density,  and 
temperature  elevation.  Furthermore  service  conditions 
demand  other  voltages  than  those  of  the  transmission  line. 
The  direct-current  transformer  or  dynamotor  is  expensive 
and  not  very  efficient. 

The  use  of  alternating  currents  is  attended  by  the  evil 
effects  of  inductance  and  capacity ;  the  operation  of  alter- 
nating-current motors  can  be  called  only  fairly  satisfactory  ; 
but  the  employment  of  the  very  satisfactory,  highly  effi- 
cient, and  moderately  priced  static  transformer,  makes 
possible  the  transmission  at  high  voltages  with  its  accom- 
panying small  currents,  small  line  wires,  and  cheap  pole 
line  construction. 

The  use  of  the  synchronous  converter  for  distribution 
purposes  in  connection  with  alternating-current  transmis- 
sion, constitutes  a  very  satisfactory  system,  and  seems  to 
best  meet  all  the  engineering  requirements. 

87.  Frequency.  —  It  is  customary  to  call  frequencies 
above  60  high,  and  those  below  60  low.  The  proper  fre- 


1 84          ALTERNATING-CURRENT   MACHINES. 

quency    for  a    transmission     and     distributing    system    is 
dependent  upon  a  number  of  variables  as  follows  :  - 

a.  High-frequency  transformers   are  smaller  and    cost 
less  than  those  of  lower  frequency.    This  is  seen  by  inspec- 
tion of  the  formula  in  article  59,  I.     For  the  same  volt- 
age and  flux  density,  the  product  of   the  iron  cross-section 
and  the  number  of  turns  varies  inversely  as  the  frequency. 
The   cross-section  of  copper  would  be  the   same  for  the 
same  capacity,  irrespective  of  the  frequency. 

b.  High-frequency    generators     may    be     constructed 
cheaper  than  those  of  low  frequency.     For  the  same  field 
multipolarity  a  high  frequency  is  associated  with  high  arm- 
ature speed,  and,  therefore,  greater  output.     On  the  other 
hand,  if  an  armature  be  run  at  the  greatest  peripheral  velo- 
city mechanically  permissible,  a  high  frequency  necessitates 
a  greater  field  multipolarity,  and,  therefore,  a  greater  cost 
and  complexity  of  construction. 

c.  High  frequencies   permit  of   the    satisfactory    oper- 
ation of  both  arc  and  incandescent  lamps.     Arcs  do  not 
operate  well  on  any  frequencies  below  40.     The  satisfac- 
tory operation  of  incandescent  lamps  depends  upon  their 
voltage  and  candle-power.       Low-voltage  lamps  have  fat 
filaments  of  large  heat  capacity  which  do  not  drop  in  tem- 
perature  so  rapidly  as  high-voltage  thin  filaments.     The 
same  is  true  of  high  candlepower  filaments.     These  lamps 
may  be  operated  satisfactorily  at  25    cycles  per  second. 
Standard  no-volt,  16  candle-power  lamps,  however,  fatigue 
the  eye  at  frequencies  under  30  cycles. 

d.  The  inductive  line  drop,  2  irfL,  varies  directly  as  the 
frequency.      Its  value  will  be   considered   later.      Being 
greater  for  high  frequencies,  it  is  then  more  liable  to  pro- 
duce poor  regulation  at  points  of  distribution. 


POWER   TRANSMISSION.  185 

e.  The  capacity  charging  current  also  varies  directly  as 
the  frequency. 

f.  The  wattless  currents  due  to  inductance  and  capacity, 
therefore,  increase  with  the  frequency,  and  thereby  lower 
the  operative  capacity  of  the  generator,  the  transformers, 
and  the  line.     They  also  lower  the  efficiency  of  operation. 

g.  High  frequencies  may  necessitate  so  high   a  field 
multipolarity  that  the  angular  speed  variation  of  the  prime 
mover  will  prevent  the  satisfactory  paralleling  of  the  gen- 
erators.    For  the  same  reason,  the  running  of  synchronous 
motors  and  of  synchronous  converters  may  be  unsatisfac- 
tory. 

//.  Induction  motors  are  best  suited  for  operation  on  low- 
frequency  circuits.  At  high  frequencies  the  speed  must 
be  high  or  the  motor  must  be  large  to  avoid  running  on  a 
low-power  factor.  The  speed  could  be  lowered  by  increas- 
ing* the  number  of  poles  ;  i.e.,  by  placing  the  poles  nearer 
to  each  other.  If  the  diameter  remained  the  same,  this 
would  result  in  an  increase  of  stator  flux  leakage,  which 
would  reduce  the  power  factor. 

88.  Voltage.  —  If  the  frequency,  the  amount  of  trans- 
mitted power,  and  the  percentage  of  power  lost  in  the  line, 
remain  constant,  the  weight  of  line  wire  will  vary  in- 
versely as  the  square  of  the  voltage  impressed  upon  the 
line.  This  depends  upon  the  fact  that  the  cross-section  of 
the  wire  is  not  determined  by  the  current  density  and  the 
limit  of  temperature  elevation,  but  by  the  permissible 
voltage  drop.  If  the  impressed  voltage  on  a  line  be 
multiplied  by  n,  the  drop  in  the  line  may  be  increased  n 
times  without  altering  the  line  loss.  For  the  line  loss  is  to 
the  total  power  given  to  the  line  as  the  drop  in  volts  is  to 


1 86 


ALTERNATING-CURRENT    MACHINES. 


the  impressed  voltage.     To  transmit  the  same  power,  but 
-th  the  previous  current  is  necessary;  and  this  current,  to 

produce  ;/  times  the  drop,   must,  therefore,  transverse  a 
resistance  ?/2  times  as  great  as  previously. 

In  transmitting  power  electrically  over  long  distances, 
the  line  cost  constitutes  a  large  part  of  the  total  invest- 


'12.5 


20 


ment.  In  such  cases  it  is  desirable  to  employ  as  high  volt- 
ages as  possible.  There  is,  however,  a  limit  to  the  voltage 
which  may  be  employed.  Mr.  Charles  F.  Scott  has  given 
some  interesting  results  of  experiments  carried  out  on  vari- 
ous pole  lines.  He  found  that  the  power  lost  through  the 
air  between  wires  increased  with  the  impressed  voltage,  and 
after  a  certain  voltage  was  reached,  increased  very  rapidly ; 
that,  with  a  given  impressed  voltage,  the  loss  decreased  as 


POWER   TRANSMISSION. 


I87 


the  distance  between  the  wires  was  increased  ;  that  atmos- 
pheric conditions,  such  as  snow,  rain,  and  humidity,  had  no 
appreciable  effect  on  the  loss  ;  that  peaked  wave-shaped 
E.M.F.'s  gave  a  greater  loss  than  flat -topped  ones;  and 
that  the  loss  decreased  as  the  diameter  of  the  wires  was 
increased.  The  relations  between  the  distance  between 
wires,  the  impresse'd  voltage,  and  the  power  loss,  is  shown 
in  Fig.  142. 


The  influence  of  the  change  of  size  of  wire  is  shown  in 
Fig.  143,  where  the  distance  between  the  wires  was  48 
inches  in  both  cases.  The  influence  of  the  size  of  the  con- 
ductor surfaces  upon  the  voltage  necessary  to  break  down 
a  dielectric  can  be  illustrated  by  the  apparatus  shown  in 
Fig.  144.  An  applied  voltage  of  sufficient  magnitude  will 
produce  a  spark  between  pointed  conductors,  although  the 


i88 


ALTERNATING-CURRENT   MACHINES. 


path  may  be  longer  than  between  those  which  are  spherical 
and  are  connected  in  parallel  with  them. 

At  high  voltages  the  leakage  is  accompanied  by  a  hissing 
sound,  and  the  wires  glow  visibly  at  night. 

The  maximum  pressure  thus  far  employed  in  practice  is 
60,000  volts.  The  Standard  Electric  Company  of  Califor- 


Q== 


Fig.  144. 

nia  uses  this  voltage  on  a  line  constructed  of  aluminium 
cable  J  inch  in  diameter,  and  made  up  of  37  strands,  the 
different  cables  being  42"  from  each  other. 

Very  long  distance  electrical  power  transmission  is  most 
economically  accomplished  by  the  employment  of  such  high 
voltages.  As  generators  cannot  be  constructed  to  give 
much  higher  than  1 3,000  volts,  and  utilization  devices  are 
also  limited  as  to  the  voltage  which  may  be  impressed  upon 
them,  step-up  and  step-down  transformers  are  necessary. 

To  produce  the  same  percentage  loss  of  power  in  a 
line  when  its  length  is  varied,  the  impressed  voltage  must 
vary  as  the  length.  The  number  of  volts  per  mile  vary 
in  practice  from  300  to  2000.  The  choice  of  voltage  is 
determined  by  balancing  the  annual  value  of  the  energy 
lost  in  the  line  against  the  interest  and  depreciation  on  the 
extra  capital  invested  necessary  to  prevent  the  loss. 

As  the  distance  of  transmission  decreases  there  arrives 


POWER   TRANSMISSION.  189 

a  point  when  step-up  transformers  can  be  dispensed  with 
and  also  some  step-down  transformers.  A  further  decrease 
of  distance  permits  of  transmission  and  distribution  with- 
out the  use  of  any  transformers. 

89.  Number  of  Phases A  comparison  of  the  weights 

of  line  wire  of  a  given  material,  necessary  to  be  used  in 
transmitting  a  given  power,  at  a  given  loss,  over  the  same 
distance,    must   be   based  upon   equal    maximum   voltages 
between  the  wires.     For  the  losses  by  leakage,  the  thick- 
ness and  cost  of  insulation,  and  perhaps  the  risk  of  danger 
to  life,  are  dependent  upon  the  maximum  value.     A  com- 
parison upon  this  basis  gives,  according  to  Steinmetz,  the 
following  results  :  — 

Relative  weights  of  line  wire  to  transmit  equal  power  over  the 
same  distance  at  the  same  loss,  with  unit  power  factor. 

2  Wires.    Single-phase 100.0 

Continuous  current     .     .     .     .        50.0 

3  Wires.    Three-phase 75.0 

Quarter-phase J45-7 

4  Wires.    Quarter-phase 100.0 

The  continuous  current  does  not  come  into  consideration 
because  of  its  voltage  limitation.  The  single-phase  and 
4-\vire  quarter-phase  system  each  requires  one-third  more 
wire  than  the  three-phase  system. 

By  use  of  the  Scott  three-phase  quarter-phase  trans- 
former the  transmission  system  may  be  three-phase,  while 
the  distribution  and  utilization  system  may  be  quarter- 
phase. 

90.  Aluminium  Line  Wire There  are  but  two  mate- 
rials  available  for   the  construction  of    long-transmission 


ALTERNATING-CURRENT   MACHINES. 

lines.  The  high  permeability  of  iron  prohibits  its  use. 
The  remaining  materials  are  copper  and  aluminium.  The 
prices  of  both  metals  vary,  and  sometimes  it  is  cheaper  to 
use  one  metal,  and  again  to  use  the  other.  A  number  of 
aluminium  lines  have  been  constructed  on  the  Pacific 
coast.  Not  all  of  them  have  proved  satisfactory.  Some 
of  them  broke  very  frequently  and  without  apparent  undue 
strain.  Experience  has  shown  that  the  troubles  were  due 
either  to  improper  alloying  or  impurity  of  the  material,  or 
to  improper  stringing  of  the  wires.  Aluminium  has  a 
large  temperature  coefficient  of  expansion.  Allowance 
should  be  made  for  this.  The  Standard  Electric  Co. 
strings  so  as  to  subject  their  aluminium  cables  to  a  strain 
of  4000  Ibs.  per  square  inch  at  20°  C.  Perrine  and  Baum 
give  the  following  data  concerning  a  line  of  commercial 
aluminium  in  which  they  were  interested  :  — 

DATA    CONCERNING    ALUMINIUM. 

Size  of  Aluminium  Wire  =  No.  i  copper. 
Resistance  of  Aluminium  Wire  =  No.  3  copper. 
Tensile  Strength  of  Aluminium  Wire  =  No.  5  copper. 
Weight  of  Aluminium  Wire  =  No.  6  copper. 

Diameter  for  the  same  conductivity  1.270  times  copper. 
Area  "     "       "  "      1.640  times  copper. 

Tensile  Strength  for  the  same  conductivity  0.629  times 

copper. 
Weight  for  the  same  conductivity  0.501  times  copper. 

91.   Line  Resistance The  resistance  of  anything  but 

very  large  lines  is  the  same  for  alternating  currents  as  for 
direct  currents.  In  the  larger  sizes,  however,  the  resist- 
ance is  greater  for  the  alternating  currents.  The  reason 


POWER  TRANSMISSION. 


for  the  increase  is  the  fact  that  the  current  density  is  not 
uniform  throughout  a  cross-section  of  the  conductor,  but 
is  greater  toward  its  outside.  The  lack  of  uniformity  of 
density  is  due  to  counter  electromotive  forces  set  up,  in 


24 


20 


(751- 

LUZ 

DC  LJ 

u.y' 
Ou 


20 


30 


70 


80 


90 


100 


40  50  60 

MILLIONS 
CIRCULAR  MILS    X  FREQUENCY 

Fig.  145- 

the  interior  of  the  wire,  by  the  varying  flux  around  the 
axis  of  the  wire  which  accompanies  the  alternations  of  the 
current.  This  phenomena  is  termed  skin  effect.  Its 
magnitude  may  be  determined  from  the  curve  in  Fig.  145. 

92.  Line  Inductance.  —  The  varying  flux,  which  is  set 
up  between  the  two-line  wires  of  a  single-phase  trans- 
mission circuit  by  the  current  flowing  in  them,  gives  rise 
to  a  self-induced  counter  E.M.F.  The  inductance  per 
unit  length  of  single  wire  is  numerically  equal  to  the  flux 
per  unit  current,  which  links  a  unit  length  of  the  line. 
To  determine  this  value  consider  a  single-phase  line,  with 
wires  of  R  cms.  radius,  strung  with  d  cms.  between  their 
centers,  and  carrying  a  current  i.  Let  a  cross-section  of 


ALTERNATING-CURRENT    MACHINES. 


-d- 


\ 


Fig.  146. 

the  line  be  represented  in  Fig.  146.     The  flux  d&lt  which 
passes  through  an  element  dr  wide  and  of  unit  length,  is 
equal  to  the  magnetomotive  force  divided    by  the  reluc- 
tance or  • 
'^  =  —r' 


dr 


Integrating  for  values  of  r  between  d  —  R  and  R 


1  =  2  /log 


and  practically 


=  2  i  log  f  —  J 


There  is  some  flux  which  surrounds 
the  axis  of  the  right-hand  wire,  and 
which  lies  inside  the  metal.  This  is  of 
appreciable  magnitude  owing  to  the 
greater  flux  density  near  the  wire. 
Represent  the  wire  by  the  circle  in 
Fig.  147,  and  suppose  that  the  current 
is  uniformly  distributed  over  the  wire. 

£ 

inside  the  circle  of  diameter  x  is  —  ^ 

iC 

motive  force,  which  it  produces,  is 


Fig.  147. 

Then  the  current 
it  and  the  magneto- 


POWER  TRANSMISSION.  193 

The   flux,  however,  which   it    produces,   links    itself  with 

x? 
but  -Hgths  of  the  wire.     The  flux  through  the  element  dx, 

which  can  be  considered  as  linking  the  circuit,  is  therefore 


Integrating  for  values  of  x  between  o  and  R, 


For  copper  or  aluminium  wires  /x  =  i.      Hence  the  total 
flux  linked  with  the  line  is 


and  the  inductance,  in  absolute  units,  being  the  flux  per 
unit  current,  is 


This  gives  by  reduction  the  inductance  in  henrys  per  wire 
per  mile  as 

L  =    80.5  +  740  log 


In  case  of  a  three-phase  line,  the  inductance  in  henrys  per 
mile  of  the  whole  circuit  is 


L  =  (139  +  1,280  log  (|- 


93.  Line  Capacity  ---  The  two  wires  of  a  single-phase 
transmission  line,  together  with  the  air  between  them, 
act  as  a  condenser.  The  wires  correspond  to  the  con- 
denser plates,  and  the  air  to  the  dielectric.  When  lines 
are  long,  or  when  the  wires  are  close  together,  the  capacity 


194 


ALTERNATING-CURRENT    MACHINES. 


DAN 

AT 


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to  HI  o 
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(S    ro  -^  too  00    O    ro  t-^  ^  \D    ^ 


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M    t 

MvO 


OOOOOOOOO 
OO^t>'0-*OtoN« 
^^  "^^  ^  M.  1°°~  "C 
to  f^vO  N  M  f5vO  O  ^O 
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vO    «v 


to  OS00    0s  ^  N    N 
NOO    toN    OOO^D 


-a 


ponding  to 
of  .90  or  .80 


f  .44  or  .60 
\f>fCE  io~ 

ncy. 
tancein  henry 
wire. 
circuit. 


- 

or 


*. 
oltag 


. 
line 
ductance  fac 
line:  7=r  2  X 
/—  Fr 
rads.  Z^Ind 
mile 
mile 
,  18" 


p 
C 


^      e 
--^      >-    -"a  S 

-vriss 
iS|^ul 

fcl^^^  u 

u^.S  «  g  g 
ffH^IIJ 


POWER   TRANSMISSION. 

CURRENT  IN  MAIN  CONDUCTOR.  VALUES  OF  T. 


195 


SYSTEM. 

PER  CENT  POWER  FACTOR. 

100 

•95 

.90 

•85 

.80 

Single-phase      
Two-phase  (4  wire)    .... 
Three-phase  (3  wire)      .     .    ,. 

I.OOO 

.500 

.576 

1.052 
.526 
.607 

1.250 

.625 
.729 

•555 
.642 

.588 
.679 

Output  in  Watts 

Current  in  main  conductors  = — - —XT. 

£. 

is  quite  appreciable.     The  capacity  of  a  two-wire  line  in 
microfarads  per  mile  of  line  is  approximately 

0.04 


where  d  is  the  distance  between  centers  of  wires,  and  R  is 
the  radius  of  the  wire,  both  being  measured  in  the  same 
units. 

Because  of  its  capacity,  a  line  which  is  unloaded  takes 
a  current  when  an  alternating  E.M.F.  is  impressed  upon 
it.  If  the  capacity  be  C  microfarads,  then  E  volts  at  a 
frequency /would  send  a  charging  current  (§21) 

1=2  7r/JSCio~6  amperes. 

94.  Line  Constants The  various  constants  of  a  trans- 
mission line  are  given  in  the  table  on  the  preceding  page. 

In  calculating  the  sizes  of  lines,  transformers,  and 
generators  of  a  transmission  system,  allowance  has  to  be 
made  for  the  various  power  factors  of  the  load  drawn  off 
at  various  points.  Induction  motors,  arc  lights,  and 
synchronous  motors  under  some  excitations  have  other 
than  unit  power-factor.  Therefore  transformers  which 
supply  them  must  have  an  excess  of  capacity  sufficient  to 


196  ALTERNATING-CURRENT   MACHINES. 

Line  loss  in  per  cent  of  power  delivered 

28  27  2625  2423   22  21   2019  18  17  16    15  14  13    12  11  10    9     8     7     6     5     4     3    2     1      0 


28  27  26  25  24  23  22  21   20  19  18  17  16  15    14  13  12  1 1   10   9     8    7     6     5     4      32     1     0 

Line  loss  in  per  cent  of  power  delivered 
Fig.  148. 


POWER   TRANSMISSION.  197 

carry  the  extra  current.  The  line,  the  step-up  trans- 
formers, and  the  generator  which  supplies  the  electrical 
energy,  must  all  have  increased  capacity.  The  prime 
mover,  which  drives  the  generator,  however,  does  not  need 
to  have  this  extra  capacity.  The  actual  current  in  all  the 
apparatus  being  larger  than  it  would  be  if  the  power- 
factor  were  unity,  'is  accompanied  by  increased  heat  losses 
at  every  point.  The  excess  of  capacity  is  needed  to  get 
rid  of  this  heat,  without  undue  elevation  of  temperature 
in  the  apparatus.  The  equivalent  impedance  of  the  loads 
and  their  equivalent  power-factor  as  affecting  the  line  can 
be  determined  as  shown  in  the  problems  of  Chapter  IV. 

95.   Weight  of  Copper In  the  curves  of  Fig.  148  are 

shown  the  relations  which  exist  between  the  transmission 
loss  of  power  in  per  cent,  the  impressed  volts  per  mile,  and 
the  weight  of  copper  per  K.W.  delivered.  The  loss  is 
expressed  as  a  percentage  of  the  power  delivered.  The 
curves  apply  to  a  three-phase  transmission  at  unit  power- 
factor.  Five  per  cent  has  been  allowed  for  sag  of  lines  be- 
tween poles.  To  determine  the  values  for  aluminium  wire, 
correct  by  the  constants  given  in  §  90. 


198  ALTERNATING-CURRENT    MACHINES. 


CHAPTER   X. 

TESTS. 

96.  Apparatus.  —  In  the  following  pages  are  given  direc- 
tions for  a  series  of  experiments  designed  to  give  the  stu- 
dent dexterity  in  handling  apparatus,  a  firmer  grasp  of  the 
phenomena  connected  with  alternating  currents,  and  a 
knowledge  of  the  methods  employed  in  testing  alternating- 
current  apparatus.  This  course  was  laid  out  for  use  in 
a  laboratory  with  but  a  moderate  amount  of  apparatus, 
and  all  this  apparatus  will  be  here  described  to  avoid  the 
necessity  of  introducing  such  descriptions  in  the  directions 
for  the  experiments. 

The  laboratory  is  supplied  with  power  from  an  Edison 
direct-current  three-wire  system  with  117  volts  on  a  side. 
The  largest  machine  is  a  7.5  K.W.  double-current  gener- 
ator, which  is  run  as  an  inverted  converter  from  the  Edi- 
son current.  This  is  a  four-pole  machine  whose  speed  can 
be  regulated  from  1200  to  1800  R.P.M.  This  gives  a 
range  on  the  alternating  end  of  40  ~  to  60  ^.  There  are 
six  slip-rings  on  the  armature,  so  connected  that  single- 
phase  current  can  be  had  from  rings  I  and  4,  quarter-phase 
from  1-4  and  2-6,  and  three-phase  from  1-3-5.  The  volt- 
age, of  course,  cannot  be  altered.  A  laboratory  not  sup- 
plied with  current  from  a  street  service  could  use  such 
a  machine,  running  it  as  a  double-current  generator  by  a 
steam  or  gas  engine.  This  would  be  more  desirable  than 
running  a  regular  alternator ;  as  frequently  direct  current, 


TESTS.  199 

as  well  as  alternating,  is  called  for  in  the  experiments.  In 
such  case,  both  frequency  and  voltage  could  be  regulated 
Besides  this,  there  is  a  500- watt,  8-pole,  125^  alternator 
belt-driven  by  a  direct-current  motor.  The  wave-shape  of 
this  machine  was  given  in  Fig.  4.  The  machine  on  which 
most  of  the  tests  are  run  is  a  double-current  generator  of 
about  i  K.W.  capacity.  This  is  a  bipolar  machine  fitted 
with  four  slip-rings  on  one  end  and  a  commutator  on  the 
other.  The  rings  are  arranged  so  that  three-phase  cur- 
rent is  obtained  from  rings  1-2-3,  and  single-phase  from 
rings  1-4.  This  machine  serves  a  multitude  of  purposes. 
It  can  be  run  as  a  direct-current  motor ;  as  a  synchronous 
motor,  either  single-phase  or  three-phase ;  as  a  converter, 
either  direct  or  inverted ;  and  when  driven  by  a  belt,  as  an 
alternator,  single-phase  or  three-phase  ;  or  as  a  direct-cur- 
rent generator,  either  shunt  wound  or  separately  excited. 
Its  speed  can  be  varied  from  1500  to  2400,  giving  fre- 
quencies of  25  ~  to  40  ~.  It  may  be  run  in  parallel 
with  the  larger  converter  when  that  is  slowed  down  to 
40  ~>.  The  equipment  of  rotating  apparatus  is  completed 
by  two  induction  motors,  one  of  one-horse  power,  the  other 
of  a  half-horse  power  capacity.  They  are  both  wound  for 
three-phase ;  but  the  smaller  is  equipped  with  a  condenser 
compensator,  as  described  in  §  67,  and  can  be  run  when 
desired  on  a  single-phase  circuit. 

The  transformer  equipment  consists  of  three  I-K.W. 
i  to  i  oil-cooled  transformers,  a  half-K.w.  i  to  2  air- 
cooled  transformer,  and  an  old  ring-wound  armature  ar- 
ranged with  taps  so  that  it  can  serve  to  transform  from  i 
to  i,  2,  3,  or  4. 

For  inductive  circuits  three  coils  are  used.  The  first, 
known  as  Coil  i,  was  described  in  §  9.  It  has  about  3000 


200          ALTERNATING-CURRENT    MACHINES. 

turns  of  No.  16  B.  &  S.  wire,  10  ohms  resistance,  and  0.2 
henrys  inductance  without  iron.  A  bundle  of  iron  wires 
1 6"  long  and  \\"  diameter  can  be  inserted  in  either  of  the 
three  coils.  Coil  2  is  in  the  shape  of  a  hollow  cylinder, 
whose  external  diameter  is  3i",  internal  diameter  2\!> ',  and 
length  3^".  It-  consists  of  about  6500  turns  of  No.  26 
B.  &  S.  wire,  with  an  indiuctance  of  o.i  henry  and  a  resist- 
ance of  60  ohms.  Coil  3  is  of  the  same  external  appear- 
ance as  Coil  2,  but  is  made  of  about  7600  turns  of  No.  25 
B.  &  S.  wire,  giving  an  inductance  of  0.141  henry  and  a 
resistance  of  60  ohms.  It  will  be  noticed  that  Coil  2  and 
Coil  3  have  the  same  resistance,  and  that  their  inductances 
are  as  i  to  V2.  Six  paraffined  paper  condensers  of  about 
two  microfarads  each,  are  used  when  condensive  circuits 
are  desired. 

The  instruments  used  are  as  follows  :  Four  hot-wire 
ammeters,  with  ranges  of  i,  3,  15,  and  20  amperes  respec- 
tively. All  but  the  first  work  across  shunts,  the  small  one, 
however,  taking  the  whole  current  through  its  hot  wire. 
These,  of  course,  are  used  for  either  alternating  or  direct 
currents.  Two  inclined  coil  ammeters  have  ranges  respec- 
tively of  5  amperes  and  50  amperes. 

There  are  three  voltmeters,  an  inclined  coil  instrument 
reading  to  65  volts  ;  a  Cardew  hot-wire  instrument,  read- 
ing to  i  50  volts;  and  a  Weston  standard  portable  voltmeter 
with  two  scales,  one  up  to  100  volts,  the  other  up  to  200. 
Any  of  these  may  be  used  on  either  alternating-  or  direct- 
current  circuits. 

For  all  the  larger  measurements  a  2.5  K.W.  indicating 
wattmeter  is  used.  For  the  finer  measurements  a  Weston 
standard  wattmeter,  having  two  scales,  is  used.  The  lower 
scale,  for  use  with  pressures  of  75  volts  or  less,  reads  up 


TESTS.  201 

to  75  watts ;  the  upper  scale,  for  use  with  pressures  of  150 
volts,  reads  up  to  150  watts.  For  this  instrument  a  shunt 
has  been  constructed,  having  a  coil  similar  to  the  current 
coil  of  wattmeter,  so  as  to  have  the  same  resistance  and 
the  same  time  constant  as  the  latter.  This  is  placed  in 
parallel  with  the  current  coil,  a  small  resistance  for  ballast 
having  first  been  placed  in  series  with  each.  The  watt- 
meter then  reads  up  to  300  watts,  and  is  as  accurate  on 
inductive  as  on  non-inductive  loads. 

Certain  direct-current  instruments  are  occasionally  used, 
principally  a  Weston  standard  portable  i5O-volt  direct- 
current  voltmeter,  and  a  similar  five-ampere  ammeter. 
These  instruments  are  used  for  convenience,  and  could  be 
dispensed  with  if  necessary. 

A  means  of  measuring  the  rate  of  rotation  of  the  various 
machines  is  essential,  and  a  portable  tachometer  is  by 
far  the  best  instrument  for  the  purpose.  Of  course,  a 
greater  accuracy  can  be  obtained  by  using  a  revolution 
counter,  and  noting  the  number  of  revolutions  in  a  con- 
siderable length  of  time ;  but  this  method  is  too  slow  to 
be  satisfactory,  and  is  useless  if  the  speed  be  fluctuating. 

To  load  a  machine  electrically,  two  lamp  boards  are 
used.  These  have  each  ten  key  sockets  arranged  in  two 
rows  between  three  wires.  Thus,  three  wires  of  a  three- 
wire  system  may  be  connected  thereto,  or  the  outside 
wires  may  be  connected  together,  and  all  the  lamps  be  put 
in  multiple ;  and  finally,  by  using  the  two  outside  wires 
only,  all  the  lamps  turned  on  in  one  row  can  be  put  in 
series  with  all  those  on  in  the  other  row.  Thus  a  wide 
range  of  resistances  can  be  obtained  by  very  small  steps, 
if  a  few  each  of  8,  16,  32,  50,  and  100  candle-power  lamps 
are  in  the  sockets. 


202  ALTERNATING-CURRENT   MACHINES. 

In  the  following  descriptions  of  experiments,  for  the 
sake  of  brevity,  the  apparatus  needed  will  not  be  named ; 
but  such  notation  will  be  used  in  the  figures,  showing  the 
arrangement  of  apparatus,  that  the  particular  apparatus 
will  be  indicated.  All  measuring  instruments  will  be 
marked  with  a  letter  indicating  their  kind,  and  a  number 
indicating  their  capacity  ;  thus  As  is  a  three-ampere  am- 
meter, W2m  is  a  2.5  K.W.  wattmeter.  In  many  cases,  the 
manner  of  drawing  will  indicate  the  apparatus,  thus  : 

Q       is  an  alternating-current  ammeter  or  voltmeter. 
Jjf^      is  a  direct-current  ammeter  or  voltmeter. 
CT       is  a  wattmeter,  the  binding  posts  of  the  current  coil 
W  being  conspicuously  large  to  avoid  confusion. 

\-^      is  a  switch  designed  to  shift  one  ammeter  out  of  cir- 
cuit and  another  in  without  interrupting  the  con- 
tinuity of  the  circuit. 
,_, —     is  a   contact-maker,   giving    a    short    contact    at    any 

desired  point  in  a  revolution. 
.^.     is  a  commutator  designed  to  change  the  direction  of 

current  flow  in  a  circuit. 
»'%S8%  is  a  lamp  board  as  described  above. 
_OQQ2_     is  an  inductive  coil, 
is  a  condenser, 
is  a  transformer,  the  numbers  indicating  the  relative 

number  of  turns. 
-*-      is  the    armature    and    field    coils    of    a    direct-current 
machine  or  the  direct-current  end  of  a  converter. 
|)|-       is  the  armature  and  field  coils  of  an  alternating-cur- 
rent   machine    or  the   alternating-current  end  of 
a  converter. 

G — ©    is  to  represent  a  belt-drive  between  two  armatures, 
is  to  represent  a  direct  connection,  or,  in  the  case  of 
a  converter,  the  two  ends  of  the  same  armature. 


TESTS.  203 

97.   Exp.  i.  Peculiarities  of  Alternating-Current  Circuits. 

—  This  experiment  consists  of  some  merely  qualitative 
observations  calculated  to  illustrate  to  the  student  the  dif- 
ference between  alternating  currents  and  the  direct  cur- 
rents he  has  hitherto  used. 

First  Part.  —  Arrange  the  apparatus  as  in  Fig.  149,  the 
lamp  being  by  way  'of  protection,  in  the  case  of  accidental 
short  circuit.  Let  x  be  first,  100*0.  P. 

the  inductive  coil  known  as  W 

Coil  i ,  second,  the  same  with    60  v/ 
the  iron  core  inserted  in  it, 


third  a  condenser  of  about  v     ' 


10  M.F.  capacity,  and  fourth 
a  5o-candle  power  lamp. 
Apply  to  these  circuits  a  Fig.  MQ- 

uniform  potential  of  about  60  volts.  Let  the  frequency 
be  successively  125  /-',  40^,  and  o  ^,  i.e.,  direct  current. 
With  each  change  note  the  ammeter  reading.  It  will  be 
observed  that  with  an  inductive  circuit  the  current  in- 
. creases  as  the  frequency  decreases,  and  that  the  maximum 
current  possible  flows  in  the  form  of  direct  current.  With 
a  condensive  circuit  the  current  decreases  as  the  frequency 
decreases,  and  is  zero  with  direct  current.  With  a  non- 
reactive  circuit,  such  as 

ii7v." rn  R,0  ' — coy     tne   50  c.  p.  lamp,  the 

D-c.« — i^ LJ 'vww  , •  '  L  n        •    '    i 

M-OI  t I  current  flow  is  mdepen- 

rOi 

^ 16  dent  of  the  frequency. 

Fig-  I5°-  Direct  current  at  60 

volts  for  this  experiment  can  be  secured,  of  course,  by 
running  a  small  dynamo  at  suitable  excitation,  but  more 
easily  from  the  1 1  /-volt  street  service  by  the  arrangement 
shown  in  Fig.  150.  The  lamps  can  be  adjusted  to  give 


2O4  ALTERNATING-CURRENT   MACHINES. 

60  volts,  and  the  rheostat  can  take  up  the  difference. 
This  adjustment  will  be,  of  course,  somewhat  different  for 
different  loads. 

Second  Part.  —  The  following  solution  is  one  of  the 
many  used  for  blue-prints,  and  has,  besides,  the  property  of 
turning  blue,  at  the  anode  only,  when  a  current  is  passed 
through  it,  if  the  anode  be  of  iron.  Mix  25  parts  (by 
weight)  of  ammonium  nitrate,  NH4  NO3,  and  12.5  parts  of 
ammonium  muriate,  NH4C1.  Dissolve  1.3  parts  of  ferri- 
cyanide  of  potassium,  K3Fe  (CN)6,  (red  prussiate  of  potash) 
in  1000  parts  of  water.  Add  the  ammonium  salts.  The 
chemicals  should  be  pure  and  the  water  distilled.  Keep 
in  a  dark  place,  and  use  within  twenty-four  hours. 

Prepare  an  insulating  handle,  Fig.  151,  with  three  piano- 
wire  projections  long  enough  to  be  elastic,  and  whose 
points  may  touch  a  plane  surface  in  a  right  line  and 

16  c.  p. 


75V. 

3  Phase  > &- 

40  «•     .       _J> 


Fig.  151. 

near  together.  Let  these  wires  be  connected  through  16 
c.  p.  lamps  respectively  to  the  terminals  of  a  three-phase 
system,  the  pressure  being  100  volts  or  less,  and  the  fre- 
quency 40  or  less.  Lay  an  uncalendered  paper  well 
moistened,  but  not  soaked,  in  the  blue-print  solution  upon 
a  metal  plate,  and  draw  the  marking-points  quickly  across 
its  surface.  Blue  marks  will  be  left  when  the  current  is 
passing  in  one  of  its  directions ;  and  these,  by  their  inter- 
rupted nature,  will  show  the  change  of  direction  in  the 
alternating  current.  Also  the  relative  displacement  or 


TESTS. 


205 


"  stagger  "  of  the  rows  of  marks  will  show 
the  phase  displacements  of  a  three-phase 
current,  as  in  Fig.  152. 

Third  Part.  —  Excite  a  i6-candle  power 
lamp  with  alternating  current  at  its  rated 
voltage  and  a  low  frequency,  say  40  ~. 
Hold  one  end  of,  a  bar  magnet  against  the 
bulb  and  in  various  positions.  The  fila- 
ment will  vibrate  synchronously  with  the 
alternations,  due  to  the  regularly  recurring 
attraction  and  repulsion  between  the  per- 
manent magnetic  field  of  the  magnet  and 
the  alternating  field  of  the  filament.  If 
this  experiment  fails  at  first,  try  varying 
the  frequency,  the  strength,  and  polarity 
of  the  magnet,  and  even  try  other  lamps. 
Often  the  filament  can  be  made  to  so  vi- 
brate as  to  touch  the  glass,  and  finally 
rupture  itself. 


£      360- 
Sb 
Q      330- 


120- 
90- 


0- 

Fig.  152. 


98.    Exp.  2.   Shape  of   E.M.F.  Wave  of 

Alternator To  perform   this  experiment 

use  is  made  of  a  balance,  as  shown  in 
Fig.  153.  It  consists  of  a  hard  graphite 
rod,  C,  of  high  resistance,  through  which 
current  is  passed  from  a  direct-current  con- 
stant potential  source,  two  16  c.  p.  lamps  being  in  series 
to  guard  against  accident  in  case  of  accidental  short  cir- 
cuit. A  rolling  contact  bears  upon  this  rod,  and  allows 
of  a  nice  adjustment  of  the  pressure  applied  to  the  testing 
circuit.  This  pressure  can  be  accurately  measured  by  the 
standard  direct-current  voltmeter  V.  In  one  branch  of 


206 


ALTERNATING-CURRENT    MACHINES. 


the  test  circuit  is  placed  the  telephone  receiver,  T.  The 
operation  is  as  follows  :  The  test  circuit,  consisting  of  the 
armature  of  the  alternator,  a  lamp  or  other  non-inductive 
resistance  for  protection,  a  contact-maker,  and  the  E.M.F. 
balance  just  described,  is  closed  for  an  instant  at  some 
point  of  the  revolution  which  corresponds  to  some  point  . 
of  the  curve  of  instantaneous  pressures.  At  such  instants 
current  will  flow  through  the  test  circuit,  causing  the 
telephone  receiver  to  click  sharply  ;  and  this  click  comes 

with  a  rapidity  corresponding 
to  the  rate  of  revolution  of 
the  contact-maker,  say  1800 
per  minute.  The  sliding  con- 
tact on  the  graphite  rod  is 
then  operated  until  the  con- 
tinuous direct  E.M.F.  is  just 
equal  and  opposite  to  the  in- 
stantaneous E.M.F.  put  forth 
by  the  alternator.  Then  there 
will  be  no  flow  of  current 
whether  the  contact-maker  be 
opened  or  closed,  and  the  re- 
ceiver will  cease  to  click.  The  voltage  can  be  read  di- 
rectly on  the  voltmeter,  thus  obviating  the  use  of  any 
reduction  constants.  This  method,  due  to  Mershon,  is 
very  delicate,  since  a  telephone  receiver  is  sensitive  to 
very  small  currents. 

To  obtain  an  E.M.F.  curve  from  an  alternator,  arrange 
the  apparatus  as  in  Fig.  154.  The  contact-maker  is  con- 
nected directly  to  the  shaft  of  the  generator,  and  is  obliged 
to  revolve  in  unison  therewith.  Run  the  alternator  at  its 
rated  speed  and  voltage.  Set  the  brush  of  contact-maker 


TESTS.  207 

at  the  desired  beginning  point,  and  balance  the  instanta- 
neous E.M.F.  by  sliding  the  balance  until  no  clicking  is 
heard  in  the  receiver.  Note  the  setting  of  the  contact- 


To  Balance 


WA/WVReo 


Fig.  154- 


maker  and  the  reading  of  the  voltmeter  in  the  balance. 
Set  the  contact-maker  ahead  by  five  electrical  degrees 
(i  mechanical  degree  =/  electrical  degrees,  where/  is  the 
number  of  pairs  of  poles),  and  repeat  as  before.  Take 
readings  thus  by  steps  of  five  degrees  throughout  one 
complete  cycle,  i.e.,  under  two  poles.  Since  the  instan- 
taneous E.M.F.  will  be  in  one  direction  during  half  a 
cycle,  but  in  the  opposite  direction  during  the  other  half, 
and  balancing  E.M.F.  is  always  in  the  same  direction, 
a  commutator  must  be  introduced  in  the  test  circuit,  as 
shown.  When  the  commutator  is  in  one  position,  the 
voltage  readings  should  be  marked  -f ,  when  in  the  other 
position,  they  should  be  marked — . 

Plot  a  curve  with  volts  as  ordinates  and  degrees  as 
abscissae.  *  Indicate  on  the  margin  of  the  curve-sheet  the 
effective  value  of  the  curve  as  obtained  from  the  alternat- 
ing-current voltmeter.  By  means  of  a  planimeter  meas- 
ure the  area  of  one  lobe  of  the  curve,  and  find  its  average 
ordinate,  by  dividing  the  area  by  the  base  line,  i.e.,  the 


208 


ALTERNATING-CURRENT    MACHINES. 


length  corresponding  to  180°.  This  may  be  done  in  inch 
and  square  inch  units,  if  the  planimeter  be  so  calibrated, 
without  reducing.  Lay  this  average  ordinate  off  on  the 
margin  also.  Divide  the  effective  value  by  the  average 
value  to  obtain  the  form-factor  (§  4)  of  the  pressure  wave. 
If  this  value  be  about  i.ii,  the  curve  is  practically  a 
sinusoid. 

99.  Exp.  3.  Shape  of  Current  Wave  of  Alternator  with 
Inductive  Load.  —  Arrange  the  apparatus  as  shown  in 
Fig.  155.  The  method  of  procedure  is  that  of  Exp.  2. 

The  instantaneous  drop  of  potential  along  a  non-induc- 
tive resistance  is  proportional  to,  and  in  phase  with,  the 
current  in  that  resistance.  Measure  the  resistance  of 


Coil  1 


To  Balance 


WWV\AMAR60 


Fig.  155- 


the  50  c.  p.  lamp  under  tJie  conditions  of  use,  since  the 
resistance  of  a  carbon  filament  varies  widely  with  the 
temperature ;  and  from  this,  and  the  values  of  instanta- 
neous pressures  observed,  calculate  the  instantaneous  cur- 
rents according  to  Ohm's  Law. 

Plot  a  curve  with  amperes  as  ordinates,  and  degrees  a§ 
abscissae. 


TESTS. 


209 


ioo.  Exp.  4.  Simultaneous  Pressure,  Current,  and 
Power  Curves  from  Alternator  with  an  Inductive  Load. — 
Arrange  apparatus  as  in  Fig.  156.  It  will  be  seen  that 
either  a  point  on  the  pressure  curve  (Exp.  2),  or  a  point 
on  the  current  curve  (Exp.  3),  can  be  taken  by  suitably 
placing  the  two-throw  switch.-  Putting  the  commutator 
in  the  main  circuit  instead  of  the  test  circuit  is  possible 
when  the  load  is  light,  does  not  affect  the  validity  of  the 
observations,  and  eliminates  a  possible  source  of  trouble 
from  bad  contacts  in  the  test  circuit.  Readings  are  to  be 


Coil  1 


D.p.,D.r. 

Switch 


Fig.  156. 

taken  every  five  degrees  through  400°,  a  little  over  one 
cycle.  Take  readings  for  both  pressure  and  current  curves 
each  time  before  moving  the  contact-maker.  This  is  better 
than  taking  a  complete  current  curve,  then  going  back  and 
taking  a  pressure  curve,  since  there  is  more  liability  to 
distortion  due  to  change  in  conditions  in  the  latter  case. 
The  voltmeter,  ammeter,  and  wattmeter  readings  should 
not  vary  during  the  test,  and  occasional  observations  should 
be  made  to  see  that  this  condition  is  complied  with.  If  it 
cannot  be,  readings  at  stated  intervals  should  be  taken, 
and  their  averages  used  in  the  subsequent  calculations, 


210 


ALTERNATING-CURRENT    MACHINES. 


Plot  three  curves  on  one  sheet,  having  degrees  as  their 
common  abscissae,  and  volts,  amperes,  and  watts  as  their 
respective  ordinates.  The  instantaneous  watts  at  any 
abscissa  equal  the  product  of  the  instantaneous  volts  and 
amperes  for  that  same  abscissa.  In  general,  a  separate 
scale  of  ordinates  will  be  required  for  each  curve.  The 
curve^will  have  the  general  relations  shown  in  Fig.  14. 

Note  the  number  of  degrees  intercepted  on  the  axis, 
between  the  pressure  curve  and  the  current  curve.  This 
is  the  angle  of  lag,  <f>,  the  cosine  of  which  is  the  power- 
factor  of  the  circuit  if  the  pressure  wave  is  sinusoidal. 
By  the  method  given  in  Exp.  2,  find  the  form-factor  of  the 
pressure  curve.  Divide  this  by  the  form-factor  of  a  true 
sinusoid,  i.e.,  i.i  i,  and  call  the  quotient  K.  Then  K  cos  <£ 
is  the  power-factor  of  the  curve,  whether  sinusoidal  or  not. 

By  means  of  a  planimeter,  measure  the  area  of  the  lobes 
of  the  power  curve,  being  careful  to  go  around  the  nega- 
tive part  in  a  counter-clockwise  direction.  Find  the  mean 
ordinate  of  this  curve  by  dividing  the  area  by  the  base 
line,  and  determine  its  value  in  watts  by  laying  off  on  the 
scale  of  ordinates  for  the  power  curve. 

Fill  out  the  following  table,  putting  in  the  last  column 
the  percentage  variation  of  the  individual  values  from  the 
average. 


How  DETERMINED. 

WATTS. 

% 

By  Wattmeter  . 

By  Planimeter 

By  E  x  /  X  K  cos  0  .     .    .,    . 

Average 

o 

TESTS.  ,      211 

The  variations  should  be  within  the  limits  of  errors  of 
instruments  and  observations,  say  2%. 

101.  Exp.  5.  Measurement  of  Self -inductance. — There 
are  various  methods  of  measuring  the  coefficient  of  self- 
induction,  .  two  of  which  are  given  here.  The  first  is 
applicable  to  any  series  circuit,  and  consists  in  the  deter- 
mination of  the  quantities  in  the  general  expression 

E 

~  \IJP + (2  7T/zy2' 

and  solution  for  L.  If  E,  /,  and  R  be  measured  respec- 
tively in  volts,  amperes,  and  ohms,  L  will  be  expressed 
in  henrys. 

(a)  Arrange  apparatus  as  shown  in  Fig.  157,  all  the 
lamps  being  turned  off.  Insert  at  x  successively  Coil  i, 
Coil  2,  and  Coil  3.  Turn 
on  lamps  until  a  good 
ammeter  deflection  is  ob- 
tained, and  note  readings  i2ov7 
of  ammeter  and  volt-  At(^ 

meter.      The    ohmic    re-  I I II?1 

sistance     must    in    each 

case  be  independently  determined  if  not  already  known. 
Take  four  sets  of  observations  with  each  coil;  with  and 
without  iron  core,  at  40  /-*,  and  60  /-*. 

Solve  for  the  inductance  in  each  case  from 


27T/ 

Without  iron  in  the  magnetic  circuit,  L  is  a  constant  of 
the  circuit,  independent  of  /  and/;  but  when  iron  is  pres- 


212 


ALTERNATING-CURRENT   MACHINES. 


ent,  it  varies  considerably  with  /  and  slightly  with  / 
The  variation  of  inductance  with  load  is  the  subject  of 
Exp.  7. 

Caution  must  be  used  in  this  experiment,  that  the  am- 
meter be  not  injured.  For  instance,  the  careless  removal 
of  an  iron  core  with  closed  circuit  may  cause  a  destructive 
increase  of  current. 

(b]  The  above  method  of  measuring  the  inductance  is 
not  applicable  to  branched  or  parallel  circuits  with  different 
time  constants,  for  the  reason  that  the  resistance  of  the 
whole  circuit,  as  measured  by  direct  current,  is  not  the 
equivalent  resistance  of  the  circuit,  as  explained  in  §  28. 
A  method  using  a  voltmeter,  ammeter,  and  wattmeter  is 
entirely  general,  is  equally  accurate,  and  does  not  require 


Fig.  158. 

the  independent  determination  of  the  resistance.  Arrange 
the  apparatus  the  same  as  in  the  first  part  of  this  experi- 
ment, with  the  addition  of  a  wattmeter,  as  shown  in 
Fig.  158.  The  method  of  procedure  is  the  same  as  be- 
fore, save  that  the  wattmeter  reading  is  also  noted  in  each 
case.  If  /,  E,  and  P  be  the  instrument  readings  in  am- 
peres, volts,  and  watts  respectively,  then  the  inductance  in 

henry s  is 

E  . 
—  sin 


TESTS. 


213 


This  equation  results  from  a  consideration   of   the  fol- 
lowing :  — 


COS         =  , 


p 

ET 


7 

Z  =       ' 


Z  sin  <j)  (see  Fig.  30). 

E. 
7Sm 


<£  (see  Fig.  ^ 

K^) 


(D 

102.   Exp.  6.     Measurement  of  Capacity When  there 

is  no  resistance  and  no  inductance  in  a  circuit  —  as  is  the 
case  with  a  condenser  —  the  general  formula 

E 


reduces  to 
hence 


/=  o>CE, 
I 


C  = 


27T/£ 

Arrange  the  apparatus  as  in  Fig.  1 59.  Let  x  be  the  six 
condensers  taken,  first  two  at  a  time,  then  three  at  a  time, 
then  all  together,  always  arranging 
them  in  parallel.  Note  the  current 
and  pressure  in  each  case,  and 
solve  for  C  by  the  above  formula. 
The  capacity  of  any  parallel  com- 
bination of  condensers  is  the  sum 
of  the  capacities  of  the  component  parts,  and  should  so  be 
shown  by  this  experiment,  If  E  and  /  be  in  volts  and 


ALTERNATING-CURRENT    MACHINES. 

amperes  respectively,  then  C  will  be  in  farads.  In  the 
report  reduce  these  results  to  microfarads  by  multiplying 
by  io(!. 

This  method  is  not  open  to  the  objections  to  the  similar 
method  of  measuring  inductance,  since  here  the  resistance 
is  practically  zero.  Yet  the  second  method,  using  the 

ittmeter,  co'uld  be  employed.     The  formula  would  be 

O)  C 

But  the  wattmeter  will  read  zero,  since  little  power  is  lost 
in  a  condenser,  so  <£  =  90°,  and 


which  is  the  same  as  deduced  from  the  general  formula. 

103.  Exp.  7.  Variation  of  Coefficient  of  Self  -Induction 
Under  Load.  —  This  experiment  may  be  performed  in  two 
parts  ;  (a)  by  varying  the  magnitude  of  the  measuring 
current,  (b)  by  using  a  constant  measuring  current,  and 
varying  the  saturation  of  the  magnetic  circuit  by  a  separate 
current  in  a  separate  winding. 

(a)  Measure  the  coefficient  of  self-induction  of  the  fine 
wire  coil  of  a  i  to  2  transformer  by  either  of  the  methods 
of  Exp.  5.  This  current  must  be  made  to  vary  by  suitable 
steps,  and  this  can  most  easily  be  done  by  applying  differ- 
ent pressures  to  the  coil.  A  wide  assortment  of  pressures 
can  be  obtained  by  using  different  brushes  of  the  converter 
supplying  the  energy,  and  the  different  steps  of  the 
i,  2,  3,  4  transformer.  Determine  the  value  of  L  for  each 


TESTS. 


of  the  conditions,  and  plot  a  curve  having  these  values  as 
ordinates,  and  the  corresponding  currents  used  in  measur- 
ing as  abscissae.  A  curve,  such  as  Fig.  160,  will  result 

.34 


w-30 

v 
or 

2 

u 


t22 


.5  1.'5  2  2.5 

MAGNETIZING- CURRENT.  AMPERES 
Fig.  160. 


3.5 


with  certain  irons  if  the  current  be  started  low  enough. 
The  sharp,.rise  of  the  curve  at  first  is  due  to  the  fact  that 
at  very  low  densities  the  permeability  increases  with  density, 
as  is  shown  in  the  curves  on  page  24,  Vol.  i. 


'NAAA/W 


Reo 


Fig.  161. 

(b]  Arrange  the  apparatus  as  in  Fig.   161.     The  meas- 
urements of  L  are  made  on  the  fine-wire  side  of  the  I  :  2 


2l6  ALTERNATING-CURRENT    MACHINES. 

transformer,  while  the  permeability  is  altered  by  direct 
current  in  the  low-pressure  side.  The  measuring  current 
should  be  kept  constant  ;  and  as  it  has  a  tendency  to  rise 
as  L  decreases,  resistance  will  have  to  be  inserted  in  the 
alternating-current  circuit  by  adjusting  R^.  Take  read- 
ings at  suitable  steps  from  zero  amperes  direct  current  to 
the  maximum  safe  temporary  ampere  capacity  of  the  coil 
in  question,  say  15  amperes  for  a  \  K.  w.  55-volt  coil. 

Calculate  the  value  of  L  for  each  of  the  steps,  and  plot 
a  curve,  using  these  values  as  ordinates  and  the  direct- 
current  magnetizing  amperes  corresponding  thereto  as 
abscissae. 

104.   Exp.   8.     Measurement   of    Mutual   Induction.  - 
Arrange  the  apparatus  as  in  Fig.  162,  the  requisite  pres- 


Iron  core  through  both 
Fig.  162. 


sure  being  secured  by  stepping  up  in  the  i  :  2  transformer. 
The  experiment  consists  of  three  parts  :  - 

(a)  Determine  the  coefficient  of   mutual  induction  be- 
tween the  two  coils  from  the  formula 


Transpose  Coil  2  and  Coil   3,  and  determine  M  again,  the 
formula  changing,  of  course,  to 


The  results    should  be  alike  if  the  same  current  flows 
in  each  case. 


TESTS.  217 

Finally  calculate  the  theoretical  value  of  M  on  the  as- 
sumption of  no  magnetic  leakage  from 

M  = 


£.,  and  LA  were  determined  in  Exp.  5.  If  the  same  meas- 
uring currents  be  used  throughout,  this  last  value  of  M 
will  be  somewhat  above  the  others,  since  there  is  some 
leakage  flux. 

(b)  With  the  arrangement  of  Fig.    162   place  the  iron 
core  with  its  end  flush  with  the  outside  of  Coil  2,  and  pro- 
jecting clear  through  Coil    3.     Move   Coil  3  by  steps  of 
2  cm.  each  from  o  to  24   cm.  from  Coil  2,  and  measure 
the  value  of  M  for  each  step.      Be  careful  that  the  iron 
core  be  not  moved  relatively  to  Coil  2. 

Plot  a  curve  with  centimeters  as  abscissae  and  values  of 
J/as  ordinates. 

(c)  Repeat  the  last,  keeping  the  iron  flush  with  Coil  3 
however,  and  moving  Coil  2.      In  this  case  the  current  in 
Coil  2  will  vary,  and  the  curve  will  be   distorted  by  the 
effects  of  load  and  saturation  as  investigated  in  Exp.  7. 

Plot  curve  of  (b)  and  that  of  (c)  on  the  same  sheet  and 
to  the  same  scale. 

Be  careful  not  to  remove  the  iron  core  entirely  from  the 
coil  that  is  carrying  the  current,  or  the  current  will  exceed 
the  capacity  of  the  ammeter. 

105.  Exp.  9.  Measurement  of  Power  in  a  Single-phase 
Inductive  Circuit.  —  There  are  various  ways  of  measuring 
power  in  alternating-current  circuits  besides  using  a  watt- 
meter, but  none  are  as  satisfactory. 

In  the  following  it  is  desired  to  measure  the  power  in 
Coil  i  :- 

(a)   By  the  three-voltmeter  method.     Arrange  the  appa- 


218 


ALTERNATING-CURRENT    MACHINES. 


1 

LgJ     i             ! 

E            J-              E           ,| 

I1 
i 

E, 

I 

Fig-  I63> 
sures  indicated.     The  power,  P,  in  the  coil  is 


ratus  as  in  Fig.  163, 
the  non-inductive 
lamp  resistance,  R, 
having  been  previ- 
ously determined. 
With  a  loo-volt  al- 
ternating-current volt- 
meter  note  the  pres- 


(b) By  the  three-ammeter  method.     Arrange  the  appa 
ratus  as  in  Fig.  164. 
A#  and  72  that  of  Alf  the  power  in  the  circuit  is 


If  /  be  the  reading  of  A5,  7X  that  of 


where  R  is  the  non-inductive  resistance  of  the  lamps, 
which  must  be  independently  determined  for  the  condi- 
tions of  operation. 


Fig.  164. 


Fig.  165. 

(c)  By  the  combined  method.  Arrange  apparatus  as  in 
Fig.  165.  If  /be  the  reading  of  A,}  72  the  reading  of  Al9 
and  E  the  reading  of  the  voltmeter,  then  the  power  in  the 
coil  is 


TESTS. 


219 


If  it  be  desired  to  compare  the  results  of  (a),  (b),  and  (<:), 
arrangements  must  be  made  so  that  the  same  difference  of 
potential  may  be  applied  to  the  terminals  of  Coil  I  in  each 
case. 

These  methods  are  rather  impractical,  and  open  to  the 
two  serious  objections  that  a  small  error  of  observation 
may  lead  to  a  serious  error  in  the  result,  and  that  the 
maximum  accuracy  can  only  be  obtained  when  about  as 
much  power  is  consumed  in  the  auxiliary  devices  as  in  the 
circuit  under  test. 

106.  Exp.  10.  Measurement  of  Power  in  Polyphase 
Circuits  by  Indicating  Wattmeters In  any  two-phase  cir- 
cuit of  four  wires  the  load  can  be  measured  by  two  watt- 
meters, one  connected  regularly  in  each  phase.  The  sum 
of  their  readings  is  the  power  in  the  circuit.  In  a  two- 
phase  four-wire  system  with  a  balanced  load,  one  of  the 
wattmeters  may  be  dispensed  with,  and  the  reading  of  the 
other  multiplied  by  two. 

In  any  two-phase,  three-wire  system  the  power  can  be 
measured  by  two  wattmeters  connected  as  in  Fig.  166. 


Load. 


Fig.   166. 


The  sum  of  the  instrument  readings  is  the  whole  power. 
In  a  two-phase,  three-wire  system,  where  all  the  load  is  con- 


22O 


ALTERNATING-CURRENT   MACHINES. 


nected  between  the  outside  wires  and  the  common  wire, 
and  none  between  the  outside  wires  themselves,  and  where 
the  load  is  balanced,  then  one  wattmeter  can  be  used  to 
measure  the  whole  power  by  connecting  its  current  coil  in 
the  common  wire  nd  its  pressure  coil  between  the  com- 
mon wire  and  one  outside  wire  first,  then  shifting  this 
connection  to  the  other  outside  wire,  as  indicated  in  Fig. 
167.  The  sum  of  the  instrument  readings  in  the  two 

positions  is  the  whole 
power.  A  wattmeter 
made  with  two  pres- 
sure coils  could  have 
one  connected  each 
way,  and  the  instru- 
Fis-*  I67-  ment  would  automat- 

ically add  the  readings,  giving  the  whole  power  directly. 
Or,  again,  a  high  non-reactive  resistance  could  be  placed 
between  the  two  outside  wires  and  the  pressure  coil  of 
the  wattmeter  connected  between  the  common  wire  and 
the  center  point  of  this  resistance.  This  requires  that  the 
wattmeter  be  recalibrated  with  half  of  this  high  resistance 
in  series  with  its  pressure  coil. 

With  the  exception  of  the  two-phase  systems,  the 
power  in  any  balanced  polyphase  system  may  be  measured 
by  one  wattmeter  whose  current  coil  is  placed  in  one  wire, 
and  whose  pressure  coil  is  connected  between  that  wire 
and  the  neutral  point.  The  instrument  reading  multiplied 
by  the  number  of  phases  gives  the  whole  power.  The 
neutral  point  may  be  on  an  extra  wire,  as  in  a  three-phase, 
four-wire  system  ;  or  may  be  artificially  constructed  by  con- 
necting the  ends  of  equal  non-reactive  resistances  together, 
and  connecting  f;he  free  ends  one  to  each  of  the  phase  wires. 


TESTS. 


221 


With  the  exception  of  the  two-phase  systems,  the 
power  in  any  ;/-phase,  w-wire  system,  irrespective  of  bal- 
ance, may  be  determined  by  the  use  of  n—  i  wattmeters. 
The  current  coils  are  connected,  one  each,  in  n—i  of  the 
wires,  and  the  pressure  coils  have  one  of  their  ends  con- 
nected to  the  respective  phase  wires,  and  their  free  ends 
all  connected  to  the  nth  wire.  The  algebraic  sum  of  the 
readings  is  the  power  in  the  whole  circuit.  Depending 
upon  the  power  factor  of  the  circuit,  some  of  the  watt- 
meters will  read  negatively,  hence  care  must  be  taken 
that  all  connections  are  made  in  the  same  sense  ;  then 
those  instruments  which  require  that  their  connections  be 
changed,  to  make  them  deflect  properly,  are  the  ones  to 
whose  readings  a  negative  sign  must  be  affixed. 

Some  specific  connections  for  indicating  wattmeters  in 
three-phase  circuits  are  shown  in  the  following  figures. 
Fig.  1 68  shows  the  connection  of  three  wattmeters  to  meas- 


Fig.  168. 


ure  the  power  in  an  unbalanced  three-phase  system.  All 
the  readings  will  be  in  the  positive  direction,  and  their 
sum  is  the  total  power.  If  a  fourth,  or  neutral  wire  be 
present,  it  should  be  used,  instead  of  creating  an  artificial 
neutral,  as  shown.  The  magnitude  of  the  equal  non- 
reactive  resistances,  used  to  secure  this  neutral  point, 


222 


ALTERNATING-CURRENT   MACHINES. 


i      . 

Balanced 
Load. 

t  t 

,                    S                        ,gQ-                     ^QJ 

.                                                                               P. 

Fig.   169. 


must  be  so  chosen  that  the  resistances  of  the  pressure-coils 
of  the  wattmeters  will  be  so  large,  compared  thereto,  as 

not  to  disturb  the  po- 
tential of  the  artificial 
neutral  point. 

Fig.  169  shows  the 
connection  of  one  watt- 
meter, so  as  to  read 
one-third  of  the  whole 
power  in  a  balanced, 
three-phase,  four-wire 
system.  If  the  system  be  three-wire,  a  neutral  point 
may  be  created  as  in  Fig.  168. 

Fig.  170  shows  the  connections  of  two  wattmeters  for 
the  determination  of  the  power  in  balanced  or  unbalanced 
three-phase  systems,  avoiding  the  necessity  of  a  neutral 
point.  The  algebraic 
sum  of  the  instru- 
ment indications  is 
the  whole  power.  If 
the  power-factor  be 
greater  than  .5,  both 
instruments  will  give 
positive  readings ;  if 
it  be  less,  one  instru-  Fig-  I7°- 

ment  will  give  a  negative  reading.  With  low  power-factors, 
such  as  given  by  a  partially  loaded  induction  motor,  it  is 
sometimes  difficult  to  determine  whether  the  smaller  read- 
ings are  negative  or  not.  If  in  doubt,  give  the  wattmeters 
a  separate  load  of  lamps  (power-factor  —  i)  and  make  the 
connections  such  that  both  instruments  deflect  properly- 
Then  connect  them  to  the  load  to  be  measured.  If  the 


4  I 

f=Z3 

Load. 

V  V 

TESTS. 


223 


terminals  of  one  instrument  have  to  be  exchanged,  then  to 
the  readings  of  that  instrument  must  be  affixed  the  nega- 
tive sign.  Fig.  171  shows  the  connections  for  one  watt- 
meter in  a  balanced  three-phase  circuit,  independent  of  a 
neutral  point.  The  free  end  of  the  pressure  coil  is  con- 
nected first  to  one  of  the  wires  opposite  that  in  which  the 
current  coil  is  con- 
nected, then  to  the 
other.  The  alge- 
braic sum  of  the 
readings  in  the  two 
positions  is  the  total 
power.  Both  read-  Fis-  Cl- 

ings will  be  positive  if  the  power-factor  is  greater  than 
.5  ;  but  one  of  them  will  be  negative  if  it  is  less. 
Hence  care  must  be  used  to  avoid  confusion  of  signs  at 
low  power-factors.  This  method,  requiring  a  two-throw 
switch  to  change  the  connection,  two  readings  of  the 
instrument,  and,  if  used  on  a  load  varying  from  high  to 
low  power-factor,  a  commutator,  to  change  the  pressure 
coil  connections,  has  little  advantage  over  the  method  of 
Fig.  1 69,  save  that  it  dispenses  with  the  necessity  for  a 
neutral  point. 

Six-phase  circuits  are  used  generally  only  between  the 
step-down  transformers  of  three-phase  transmission  sys- 
tems and  the  alternating-current  ends  of  rotary  conver- 
ters ;  hence  they  are  always  balanced.  They  can  then  be 
measured  by  the  method  of  Fig.  169,  where  a  neutral  is 
employed,  or  the  three  alternate  wires  may  be  considered 
a  three-phase  system,  the  method  of  Fig.  171  employed, 
and  the  three-phase  power  thus  determined  multiplied  by 
2  to  give  the  total  power.  If  the  circuit  should  be  unbal- 


224  ALTERNATING-CURRENT    MACHINES. 

anced,  five  instruments  would  be  necessary,  as  stated  earlier 
in  this  section. 

The  student  is  expected  to  construct  circuits  according 
to  the  various  figures  just  given,  and  convince  himself 
that  the  wattmeters  do  give  the  true  power.  If  the  load 
be  of  lamps,  the  power  in  each  may  be  measured  by  a  volt- 
meter and  ammeter  used  at  their  terminals  ;  then  by  con- 
necting in  star  and  in  delta,  balanced,  and  unbalanced,  the 
accuracy  of  the  wattmeter  indications  can  be  checked. 

In  following  Fig.  166  or  Fig.  167,  it  should  be  remem- 
bered that  a  two-phase  current  cannot  be  secured  from  an 
armature  with  a  mesh  winding,  such  as  a  rotary  converter 
must  have  ;  and  that  any  attempt  to  make  a  two-phase, 
three-wire  system  put  of  a  quarter-phase  system  will  be 
disastrous.  To  get  two-phase  current  from  such  a  ma- 
chine, the  quarter-phase  current  must  be  passed  through 
the  primaries  of  two  similar  transformers,  two  opposite 
wires  going  to  one,  the  other  two  to  the  other.  The 
transformer  secondaries  will  then  deliver  true  two-phase 
current,  and  the  circuits  may  be  united  in  a  three-wire 
system. 

107.  Exp.  ii.  Calculation  and  Measurement  of  the  Re- 
sulting Impedance  of  a  Number  of  Impedances  in  Series.  — 

Arrange  apparatus  as  shown  in  Fig.  172.  Determine  the 
impedance  Z  of  the  whole  circuit  from  the  readings  of  the 
voltmeter  and  ammeter. 


Independently  determine  the  ohmic  resistance  of  the  cir- 
cuit with  the  same  current  flowing.  The  magnitude  of 
the  current  affects  the  resistance  of  the  lamp. 


TESTS,  225 

Solve  for  the  reactance,  X  =  2  irfL,  from  the  equation, 

Z  =  V^?2  +  «?!?. 

Determine  the  angle  of  lag  <j>  from 

reactance 


tan  <    = 


resistance 


Determine  the  reactance,  resistance,  and  impedance  of 
Coil  i,  Coil  2,  and  the  lamp,  individually.  The  first  two 
can  be  derived  from  the  data  of  Exp.  5  without  further 
measurements. 

Graphically  determine  the  total  reactance,  resistance,  im- 
pedance, and  angle  of  lag  by  combining  the  individual  parts 
in  a  parallelogram  of  impedances  as  described  in  §  26.  A 
convenient  scale  for  the  actual  plotting  for  a  drawing-board 
25"X3o"  is  2  ohms=i  cm. 

Make  a  report  in  the  form  of  a  table  such  as  .the  follow- 
ing. The  variation,  with  careful  work  and  good  instru- 
ments, should  not  exceed  2%. 


DETERMINED 

% 
VARIATION. 

GRAPHICALLY. 

EXPERIMEN- 
TALLY. 

K     ... 

X    .     .     . 

Z     .     .     . 

0     ... 

226  ALTERNATING-CURRENT    MACHINES. 

100  c.  p. 


120V. 


Fig.  172. 

108.   Exp.  12.     Calculation  and  Measurement  of  the  Re- 
sulting Impedance  of  a  Number  of  Impedances  in  Parallel. 

—  Use  the  same  impedances  as  in  the  last  experiment,  but 
arranged  as  in  Fig.  173.  As  before  stated,  the  voltmeter- 
ammeter-resistance  method  of  solving  inductive  circuits  is 
inapplicable  to  branched  circuits  ;  so  the  wattmeter  must 
be  used  as  shown. 

Determine  the  equiva-  (^=$\VWQ 

lent  impedance  from 

Coil  1. 


Coil  2. 


60V. 


100  c.  p. 


Determine    the     angle 
of  lag  in  the  main  circuit 

by 

cos  <£  =  —  - 

Determine  the  equiva- 
lent resistance,  R  (which 
is  not  the  actual  resistance  of  the  parallel  arrangement), 
from 

R  =  Z  cos  <j>. 

Determine  the  equivalent  reactance  from 
X  —  Z  sin  d>. 


Fig.     173. 


TESTS. 


227 


All  the  constants  of  Coil  i  and  Coil  2  are  known ;  but 
the  resistance  of  the  lamp  had  better  be  redetermined  for 
the  particular  current  used  in  it. 

Combine  the  admittances  of  these  parts  of  the  branched 
circuit  into  a  polygon  of  admittances  according  to  §  28. 
Take  the  reciprocal  of  the  resulting  admittance,  —  that 
is,  the  equivalent  impedance,  —  and  resolve  it  into  its  com- 
ponent parts  of  equivalent  reactance  and  equivalent  resist- 
ance. The  actual  plotting  may  be  done  on  2%"  x  30" 
drawing-board  to  the  scales  2  ohms  =  I  cm.  and  I  unit  of 
admittance  =  1000  cm. 

Make  a  report  in  the  form  of  a  table  such  as  is  used  in 
the  last  experiment.  The  variation  should  not  exceed  3%. 

109.  Exp.  13.  Calculation  and  Measurement  of  Result- 
ing Impedance  of  any  Series-Parallel  or  Parallel-Series 
Arrangement  of  a  Number  of  Impedances.  —  Arrange  the 
apparatus  as  in  Fig.  174,  or  according  to  any  other  scheme 
if  it  be  desired  to  vary  the  experiment. 


70V. 
60  ~ 


Fig.   174 


Determine  the   values  of   the    resulting    or    equivalent 
Ry  Xy  Zy  and  <£,  as  in  Exp.  1 2. 


228  ALTERNATING-CURRENT   MACHINES. 

Also  determine  the  same  quantities  for  the  individual 
parts  of  the  circuit  under  the  conditions  of  use  if  they  be 
not  already  known. 

In  the  graphic  determination  pursue  the  following 
steps  :  — 

1.  Find  the  equivalent  impedance  of  Coil  3,  and  the 

100  C.P.  lamp,  calling  it  M. 

2.  Find    the   equivalent    impedance   of   the    50   c.  P. 

lamp,  and  J/,  calling  it  JV. 

3.  Find   the   equivalent    impedance   of    Coil    i,   and 

Coil  2,  calling  it  P. 

4.  Find  the  equivalent  impedance  of  P  and  N.     This 

will  be  the  required  impedance  of  the  whole 
circuji,  and  should  be  resolved  into  its  com- 
ponent parts  of  equivalent  resistance  and 
equivalent  reactance.  Measure  <£,  the  angle 
between  the  impedance  and  the  resistance. 

Make  a  report  in  the  form  cf  a  table  as  in  the  two  pre- 
ceding experiments.  The  variation  of  the  determinations 
by  the  two  method  should  not  exceed  3$. 

no.  Exp.  14.  Efficiency  and  Regulation  of  a  Trans- 
former.—  Arrange  the  apparatus  as  in  Fig.  175.  A 
two-throw  switch  allows  the  same  voltmeter  to  read  either 
primary  or  secondary  pressure.  The  ammeter  As  may  be 
used  on  the  lower  readings.  The  transformer  used  is 
the  %  K.  w.  i  to  2,  stepping  up  from  about  58  volts  to  1 16, 
its  rated  range.  It  is  operated  at  its  rated  frequency, 
60  ~ . 

Increase  the  load  from  o  to  i  K.  w.  (100%  overload)  by 
suitable  steps.  At  each  step  take  readings  of  the  primary 
volts,  primary  watts  ;  secondary  volts  and  secondary  am- 


TESTS. 


229 


peres.     Since  the  load  is  non-inductive,  the  product  of  the 
secondary  volts  and  amperes  gives  the  secondary  watts. 

Determine  the  efficiency  and   the    regulation,  both  in 
per  cent,  for  each  set  of  readings  from 

.     ~  .  watts  secondary 

%  efficiency    =  -          — : J-  x  100. 

watts  primary 


T  volts  prim.  —  volts  sec. 
%  regulation  =  - 
'       s  full-load  sec.  volts 


X  100. 


Fig.  175. 

Plot  two  curves  on  the  same  sheet,  having  as  their 
common  abscissae  both  watts  and  per  cent  of  full-load 
secondary,  and  as  their  respective  ordinates  per  cent  effi- 
ciency and  1 00%  —  per  cent  regulation. 

in.  Exp.  15.  Determination  of  Load  Losses  in  a 

Transformer The  core  losses  are  usually  considered 

independent  of  the  load,  while  all  those  that  vary  with  the 
load  are  called  the  load  losses.  Their  chief  component  is, 
of  course,  the  I2 R  loss  in  the  copper,  but  there  may  be 
some  eddy  current  and  local  hysteresis  losses  that  vary 
with  the  load,  and  a  determination  of  them  all  is  made  as 
follows  :  — 

Arrange  the  apparatus  as  in  Fig.    176.      The   i   to   2 


230 


ALTERNATING-CURRENT    MACHINES. 


Fig.  176. 


0.5  K.  w.  transformer  is  used  with  its  low-tension  side 
short-circuited.  There  will  be  but  a  small  pressure  gener- 
ated therein,  and  its  current  will  demagnetize  the  core 

almost  entirely  ;  hence 
all  the  losses  measured 
may  be  considered  as 
load,  not  core  losses. 
Care  must,  of  course, 
be  taken  to  control  the 
amount  of  current  pass- 
ing through  the  trans- 
former. 

Adjust  the  lamps  so 
that  about  100%  overload  current,  10  amperes,  is  shown 
by  Au.  Read  the  ammeter  and  the  wattmeter.  Reduce 
the  current  by  a  suitable  amount,  and  read  again.  So 
continue  down  to  zero  amperes,  substituting  A^  for  A}. 
when  the  readings  on  the  latter  become  unsatisfac- 
tory. 

Plot  a  curve  with  load  in  amperes  as  abscissae  and  load 
loss  in  watts  as  ordinates. 

Take  care  that  the  wire  short-circuiting  the  low- 
pressure  coil  is  of  low  resistance,  and  has  good  contacts. 
Note  also  that  the  pressure  leads  from  the  wattmeter 
should  go  direct  to  the  terminals  of  the  transformer,  as, 
in  general,  the  resistance  of  the  wires  leading  to  it  is 
not  negligible  in  comparison  with  the  resistance  of  the 
coil  itself. 

If  the  current  exceeds  the  ampere  capacity  of  the  watt- 
meter, it  is  advisable  to  put  in  a  single-pole  switch  to 
short-circuit  the  current  coil  at  all  times  save  when  a 
reading  is  being  taken. 


TESTS.  231 

112.  Exp.  16.  Determination  of  Core  Losses  of  a 
Transformer,  and  Construction  of  an  Efficiency  Curve.  - 

The  core  losses,  hysteresis  chiefly,  are  constant  at  all 
loads.  Hence,  the  energy  supplied  to  a  transformer  when 
its  secondary  is  open-circuited,  is  practically  a  measure  of 
these  losses. 

Connect  a  wattmeter  in  the  primary  circuit  of  the 
i  K.  w.  transformer  when  its  primary  is  supplied  with 
pressure  at  its  rated  voltage  and  frequency,  and  its  sec- 
ondary is  open-circuited. 

The  wattmeter  reading  is  the  core  loss. 

From  a  knowledge  of  the  core  loss  and  the  load  losses 
at  various  loads,  construct  an  efficiency  curve  for  various 
loads  from  o  to  100%  overload  (secondary),  the  efficiency 
in  per  cent  at  any  load  Pl  being 

-  P\ 

X  ioo, 


where  Pl  and  Pc  are  the  load  and  core  losses  respectively 
at  the  load  Pr 

This  curve  should  be  similar  to  the  efficiency  curve 
found  in  Exp.  14. 

113.  Ex.  17  and  18.  Simultaneous  Pressure  and  Cur- 
rent Curves  from  Primary  and  Secondary  of  a  Trans- 
former. —  It  is  desired  to  get  these  curves  for  two  con- 
ditions. First,  Exp.  17,  with  a  full  non-inductive  load, 
and  second,  Exp.  18,  with  an  equal  (in  amperes)  very 
inductive  load. 

Arrange  apparatus  as  in  Fig.  177.  For  the  non- 
inductive  load,  lamps  are  suitable,  for  the  inductive  load 
the  primaries  of  unloaded  transformers  are  good ;  and  to 
get  a  nice  adjustment  Coil  i  can  be  put  into  circuit,  and 


232 


ALTERNATING-CURRENT    MACHINES. 


the  current  '"n  it  adjusted  by  moving  its  iron  core  in  or 
out. 

It  might  be  here  remarked  that  if  the  transformer  is 
supplied  with  current  from  a  rotary  converter,  the  E.M.F. 
balance  described  in  Exp.  2  cannot  use  direct  current  from 
the  same  source  as  that  which  runs  the  converter,  even 
though  they  be  put  on  opposite  sides  of  a  three-wire 
system.  A  separate  source  of  direct  E.M.F.  must  in  such 
case  be  supplied  for  the  balance,  either  from  a  separate 
direct-current  generator  or  from  a  sufficient  number  of 


To  Load 


and  Balance 


Fig.   177. 


cells  of  storage  battery.  If,  however,  the  alternating  cur- 
rent be  passed  through  another  transformer  before  being 
applied  to  the  one  under  test,  this  trouble  does  not  arise ; 
but  the  introduction  of  the  second  transformer  has  a  dis- 
turbing effect  on  the  wave-shape. 

It  will  be  seen  that  the  apparatus  is  merely  an  elabora- 
tion of  that  in  Exp.  4,  a  four-way  double-pole  switch  being 
used  instead  of  the  two-throw  switch  of  the  former  experi- 
ment. This  switch  is  conveniently  made  of  mercury  cups 
in  a  block  of  wood.  For  the  i  K.  w.  transformer  at  58  to 
1 1 6  volts,  the  reading  on  A15  should  be  about  4.4  amperes. 


TESTS.  233 

Suitable  values  for  the  non-ind active  resistances  are,  Rp  — 
2.2  ohms,  Rg  =  4.4  ohms.  These  must  be  able  to  carry 
the  currents  without  overheating,  and  must  not  be  allowed 
to  change  their  resistance  due  to  change  of  temperature. 

Proceed  as  directed  in  Exp.  4,  taking  readings  every 
twelve  electrical  degrees  throughout  half  a  cycle.  Take 
all  four  readings  before  moving  the  contact-maker. 

Plot  the  four  curves  of  Exp.  1 7  on  one  paper,  and  those 
of  Exp.  1 8  on  another.  In  each  case  degrees  will  be 
the  common  abscissae.  Both  pressure  curves  of  either 
experiment  must  be  plotted  to  one  scale  of  ordinates,  both 
current  curves  to  another.  Careful  work  will  show  a 
phase  difference  slightly  less  than  180°  between  the 
primary  and  secondary  pressures  and  currents,  particularly 
in  Exp.  1 8. 

114.  Exp.  19.  Calculation  and  Measurement  of  the 
Mutual-inductance  of  Transformer  Coils  at  No  Load.  - 

(a)  Measure  the    self-inductance    of   the   primary   and 
of  the  secondary  coils  by  the  method  of  Ex.  5,  first  part, 
the  coil  not  under  test  being  left  open-circuited. 

(b)  Do  the  same  by  the  method  of  Exp.  5,  second  part. 
The  results  in  the  two  cases  should  be  alike  if  attention  is 
paid   to  the  following   point.      In  measuring  the   primary 
inductance,  apply  the  rated  voltage  so  that  it   will    send 
the  charging  current.     In  measuring  the  secondary,  adjust 
the  impressed  voltage  so  that  just  such  a  current  will  flow 
as  will  give  the    same  ampere-turns   in  the  secondary  as 
there  were  in  the  primary  when  it  was  being  measured. 
If  this  precaution  be  not  taken,  the  results  will  be  changed 
by  the  effects  of  varying  load,  according  to  Exp.  7. 

(c)  Using  the  average  of  the  values  of  Lf  and  L,  as 


234  ALTERNATING-CURRENT    MACHINES. 

found  in  (a)  and  (b),  calculate  the  value  of  M  on  the  sup- 
position of  no  magnetic  leakage,  from 

M  =  VZ.ZP. 

(d)  Measure  the  mutual  induction  by  the  method  of 
Exp.  8  a,  taking  care  that  the  ampere  turns  are  the  same 
in  each  case,  and  the  same  as  were  used  in  (a)  and  (b). 

This  last  result  may  be  slightly  less  than  that  arrived  at 
in  (c)  because  of  magnetic  leakage. 

Care  should  be  taken  throughout  that  the  frequency  be 
kept  constant. 

115.  Exp  20.  Practice  in  Three-Phase  Transformer 
Connections  ----  Three  similar  i  to  i  transformers  may  be 
conveniently  used  for  this  experiment.  The  student  may 
have  to  exercise  some  ingenuity  in  determining  the  direc- 
tion of  winding  in  the  coils.  When  each  of  the  following 
connections  has  been  made,  excite  the  primaries  by  a  three- 
phase  current,  and  measure  the  pressure  between  each  of 
the  secondary  wires,  seeing  that  all  three  sides  have  the 
same  and  the  expected  voltage. 

(a)  Connect  both  primaries  and  secondaries  in  Y,  as 
shown  in  Fig.  88.  See  that  Et  —  Ep.  Then  make  the 
secondary  a  three-phase,  four-wire  system,  and  see  that  the 
voltage  between  any  outside  wire  and  the  middle  wire  is 


(b)  Connect  both  primaries  and  secondaries  in  A,  Fig.  87, 
and  see  that  Et  =  Ep.     Disconnect  one  transformer  from 
the  circuits,  and  observe  that  the  three-phase  pressure  is 
still  maintained  in  the  secondary. 

(c)  Connect  the  primaries  in  Y,  and  the  secondaries  in 

yET 

A,  as  in  Fig.  90.     Observe  that  Et  =  —£. 


TESTS.  235 

(d)  Connect  the  primaries  in  A,  and  the  secondaries  in 
Y,  as  in  Fig.  89,  with  a  four-wire  secondary  system.  Ob- 
serve that,  with  reference  to  the  outside  wires,  Et  =  V3  Ep) 
while  considering  any  outside  wire  and  the  middle  wire, 
E.  =  E, 

116.  Exp.  21.     External    Characteristic    of    an    Alter- 
nator  Run    the   alternator   at  normal    speed  and   field 

excitation,  both  being  kept  constant  during  the  experiment. 
Arrange  a  variable  non-inductive  load  —  lamps  preferably  - 
so  that  readings  can  be  taken  from  o  load  to  50%  overload 
at   suitable   intervals.     At    each   step  note  the    armature 
current  and  the  terminal  pressure. 

Plot  a  curve  with  currents  as  abscissae,  and  pressures  as 
ordinates. 

117.  Exp.  22.     Field  Compounding  Curve  of   an  Alter- 
nator  Run   the   alternator    at    constant    rated    speed. 

Arrange  a  variable  non-inductive  load  of  lamps,  ranging 
by  suitable  steps  from  o  load  to  50%   overload  ;  at  each 
step  adjust  the  field  current,  so  that  the  rated  terminal 
voltage  is  maintained.     Take  simultaneous  readings  of  field- 
current  and  armature  current. 

Plot  a  curve  with  armature  currents  as  ordinates,  and 
field  currents  as  abscissae. 

Note  that  the  speed  must  be  kept  constant,  that  the 
terminal  pressure  must  be  kept  constant,  and  that  read- 
ings should  be  taken  only  with  ascending  values  of  field 
currents,  as  magnetic  retentivity  will  distort  the  curve 
somewhat  if  the  field  current  is  run  too  high,  and  then 
brought  down  to  the  required  point. 

118.  Exp.  23.     No-Load  Saturation  Curve  of  an  Alter- 
nator. —  Run  the  alternator  at  constant  rated  speed,  and 


236 


ALTERNATING-CURRENT   MACHINES. 


excite  the  fields  from  zero  up  to  full  excitation,  taking,  at 
suitable  intervals,  readings  of  the  field  current,  and  the 
no-load  armature  voltage.  Repeat,  carrying  the  excitation 
from  full  excitation  down  to  zero. 

Plot  the  two  curves  on  one  sheet,  using  field  currents  as 
abscissae,  and  terminal  pressures  as  ordinates.  The  two 
curves  will  not  exactly  coincide,  because  of  the  magnetic 
retentivity  of  the  iron. 

Care  must  be  taken  always  to  adjust  the  field  current  by 
increasing  from  a  lower  value  to  a  higher  when  taking  the 
ascending  curve  ;  and  by  decreasing  from  a  higher  value  to 
a  lower  when  taking  the  descending  curve. 

119.  Exp.  24.  Full-Load  Saturation  Curve  of  an  Al- 
ternator  Arrange  apparatus  as  in  Fig.  178.  The  alter- 
nator is  a  IK.  w.  80  volt,  single-phase  machine.  The 
machine  is  given  a  non-inductive  load  of  lamps,  and  a 
heavy  current  rheostat  which  has  zero  resistance  on  the 


Fig.   I78. 


last  point.  Run  the  alternator  at  its  rated  speed.  Make 
the  resistance  of  the  external  circuit  zero  —  i.e.,  short-circuit 
it  through  the  ammeter.  Adjust  the  field  rheostat  to  its 
maximum  resistance,  and  close  the  field  switch.  Increase 
the  excitation  by  manipulating  the  field  rheostat  until  the 
rated  full-load  current  is  flowing  in  the  external  circuit,  as 


TESTS. 


237 


shown  onA15.  Take  readings  of  the  field  amperes,  and  the 
terminal  volts,  the  latter  being  zero  at  this  step.  Increase 
the  resistance  of  the  armature  circuit  by  a  suitable  amount, 
and  readjust  the  excitation  till  the  rated  full-load  current  is 
again  flowing  in  the  armature,  and  take  readings  of  field 
current  and  terminal  voltage.  Repeat  at  suitable  steps 
until  full  field  excitation  is  obtained. 

Plot  a  curve  on  the  same  paper,  and  to  the  same  scale  as 
that  of  Exp.  23,  using  field  currents  as  abscissae,  and 
terminal  volts  as  ordinates. 

Take  heed  that  readings  are  always  taken  with  ascend- 
ing values  of  field  current,  and  when  the  ammeter  in  the 
armature  circuit  shows  rated  full-load  current.  The  speed 
must  be  kept  constant. 

120.  Exp.  25.  Synchronous  Impedance  of  an  Alterna- 
tor.—  As  stated  in  §  38,  the  synchronous  impedance  of  an 
alternator  varies  somewhat  with  the  load,  but  is  practically 
constant  at  all  excitations  ;  hence  its  determination  is  easily 
accomplished  in  the  following  manner  :  — 

Arrange  the  apparatus  as  shown  in  Fig.  1 79.     Run  the 
alternator  at  its  rated  speed.     By  means  of  the  field  rheo- 
stat cut  the  excita- 
tion down  to  a  mini- 
mum.    Short-circuit 
the       armature 
through     an    am- 
meter   and     switch 
as    shown.     Adjust 
excitation     so    that 
the  ammeter  shows  about  -|  full-load  current,  and  note  the 
ammeter  reading.     Open  the  switch  in  the  armature  cir- 


Fig.  179. 


238  ALTERNATING-CURRENT   MACHINES. 

cuit  and  note  the  terminal  volts.  Close  the  switch,  read- 
just excitation  so  that  the  load  is  increased  by  a  suitable 
amount,  and  repeat  the  readings.  Repeat  until  a  limit  is 
reached,  either  because  full  field  excitation  has  been  ob- 
tained, or  because  the  machine  is  being  too  severely  over- 
loaded. Which  of  these  two  conditions  arises  first  depends 
upon  the  synchronous  impedance  of  the  machine. 

Calculate  the  synchronous  impedance  for  each  set  of 
readings  from 

open  circuit  voltage 

Syn.  Imp.  =  -,—   — : —         — , 

short  circuit  current 

when  the  readings  are  for  the  same  excitation  and  speed. 

Plot  a  curve  with  armature  currents  as  abscissae,  and  the 
values  of  the  synchronous  impedance  as  ordinates. 

Particular  attention  should  be  paid  that  the  speed  be 
kept  constant,  as  it  is  liable  to  rise  on  throwing  off  the 
load. 

121.  Exp.  26.  Core  loss  of  an  Alternator The  core 

loss  of  any  armature  is  determined  by  measuring  the  dif- 
ference in  power  required  to  run  it  with  and  without  field 
excitation.  With  an  alternator  this  is  most  easily  done  by 
running  the  armature  by  a  rated  motor  and  observing  the 
power  input  thereto.  It  is  desirable  to  have  the  quantity 
sought  as  large  as  possible  in  comparison  with  the  quanti- 
ties observed  ;  hence  the  rated  motor  used  should  be  as 
small  as  is  practicable. 

The  alternator  must  be  driven  at  its  rated  speed,  and  the 
pulleys  so  proportioned  that  the  motor  will  run  at  its  rated 
speed  also ;  or  else  a  special  efficiency  curve  of  the  motor 
must  be  obtained  for  the  speed  at  which  it  will  be  required 
to  run. 


TESTS.  239 

A  wattmeter  placed  in  the  motor  circuit  will  indicate 
the  power  input  thereto ;  or,  if  it  be  a  direct-current  motor, 
a  voltmeter  and  ammeter  can  be  used. 

Let  A  =  watts  input  to  motor  when  the  alternator  field 

is  not  excited. 

and  m  =  efficiency  of  motor  at  this  input. 
Let  £= watts  input  to  motor  when  the  alternator  fields 

are  fully  excited, 
and  «  =  efficiency  of  motor  at  this  input. 

Then  the  core  loss  in  watts  is 

Pc  =  Bn  -  Am. 

It  is  well  to  repeat  the  measurements  a  number  of  times 
and  average  the  results. 

Since  the  losses  in  shafting  and  belting  are  practically 
the  same  at  all  loads,  these  do  not  affect  the  accuracy  of 
the  results. 

122.  Exp.  27.  Complete  Test  of  a  i-H.P.  Three-Phaso 

Induction  Motor As  a  test  of  the  motor  performance 

solely,  the  voltage  at  the  motor  terminals  should  be  kept 
constant  throughout  the  test.  This  may  easily  be  accom- 
plished if  the  motor  is  run  from  a  separate  alternator.  If, 
however,  it  is  run  from  an  inverted  converter,  and  particu- 
larly if  the  desired  voltage  has  to  be  obtained  by  transfor- 
mation, there  will  be  a  slight  drop  of  voltage  as  the  load 
increases. 

Since  the  power-factor  in  this  test  will  run  from  very 
low  to  about  80%,  the  method  of  measuring  three-phase 
power  shown  in  Fig.  169  will  be  used,  as  it  requires  but 
one  instrument  reading,  and  leads  to  no  uncertainty  as  to 


240 


ALTERNATING-CURRENT    MACHINES. 


algebraic  signs.  The  apparatus  used  is  simply  an  amme- 
ter, a  voltmeter,  and  a  wattmeter  connected  into  the  motor 
circuit.  Fig.  180  shows  the  arrangement,  the  two  watt- 
meters not  being  used  at  once,  but  being  alternative, —  one 
for  high  readings,  the  other  for  low,  —  thus  securing  a 
greater  accuracy  over  a  wide  range.  The  motor  when 
stalled  takes  about  18  amperes;  so  this  is  its  momentary 


Fig.  180. 

starting  current.     Care  must  be  taken  in  starting  up  that 

the  measuring  instruments  are  not  injured  by  such  a  flow 

of    current.       The    larger    wattmeter    has   a    capacity    of 

2.5   K.  w.,  and  a  2 5 -ampere  limit ;  the  smaller  a  capacity 

for  300  watts,  and  a   5-ampere  limit.     Either  of  these,  as 

well  as  the  voltmeter,  will  stand  the  rated  motor  pressure, 

-  1 10  volts. 

The  power  output  of  the  motor  is  absorbed  in  a  strap 


TESTS. 


241 


brake,  as  shown  in  Fig.  181.  With  a  4.5"  pulley  at  1800 
revolutions  the  spring  balances  should  have  ranges  of 
about  30  Ibs.  and  4  Ibs.  respectively. 

The  motor  must  be  supplied  with  current  at  its  rated 
voltage  and  frequency,  and  the  frequency  must  be  kept 
constant  throughout  the  experiment. 

Observation%-^-1ak&  readings  at  suitable 
intervals,  —  say  steps  of  4  Ibs.  each  on  the 
larger  scale,  —  from  no  load  to  the  stalling 
of  the  motor.  Do  not  leave  the  motor 
stalled,  as  it  overloads  the  instruments. 

At  each  step  take  readings  of  the  watt- 
meter, ammeter,  voltmeter,  both  spring 
balances,  and  the  speed  of  the  motor. 

Repeat  the  experiment  three  times  with 
fifteen-minute  intervals  between  the  repeti- 
tions.    The  scale  readings,  P,  will  be  the          Flg*  l8lt 
same,  at  any  one  step,  for  all  three  trials,  and  the  other 
values  can  be  averaged   to    partially  eliminate  errors  of 
observation. 

Calculations.  --  Using  the  average  values  of  the  three 
readings  at  any  one  step,  fill  out  the  following  table  :  — 


- 

2 

3 

4 

5 

6 

7 

8 

9 

h 

% 

H 

H 

* 

* 

WATTS  OUTPU 

WATTS  INPUT. 

VOLTS  AT 
TEKMINA 

AMPERES 
PER  PHA 

VOLT-AMPERES 
INP 

1 
EC, 

A 

1 

EFFICIENCY  %. 

APPARENT 
EFFICIENCY 

0, 

n 

in 

242  ALTERNATING-CURRENT   MACHINES. 


(1)  Watts  output  =  -  746, 

33,000 

where 

d  =  diameter  of  pulley  in  inches. 
V  =  revolutions  per  minute. 
(P  —  Pr)  =  difference  in  scale  readings  in  pounds. 

(2)  Watts  input  =  3  X  wattmeter  reading. 

(3)  Volts  at  terminals  =  voltmeter  reading. 

(4)  Amperes  per  phase  =  ammeter  reading. 

(5)  Volt-amperes  input  =  V3  X  volts  at  terminals  X  amperes 

per  phase. 

\Vatts  input 

(6)  Power-factor  %  =  ^r  —.  --  x  100. 

Volt-amperes  input 

/  \    i-rc  •  ~/        Watts  output 

(7)  Efficiency  %  =  —  --  :  —    -  X  100. 

Watts  input 

Watts  output 

(8)  Apparent  efficiency  =  Tr  ~  --  X  100. 

Volt-amperes  input 


(9)    Slip  #= 

where 

V  =  revolutions  per  minute, 

/  =  frequency, 

/  =  number  of  pairs  of  poles. 

Plotting  of  Curves.  —  Plot  eight  curves  on  one  paper. 
All  the  curves  will  have  watts  output  as  abscissae.  The 
points  of  25%,  50%,  7$%,  100%,  and  125%  of  full  load 
should  also  be  indicated. 

The  ordinates  for  the  first  seven  curves  are  taken  from 
columns  2  to  8  respectively. 


TESTS. 


243 


The  ordinates  for  the  last  curve  are  found  by  subtract- 
ing the  per  cent  slip  from  100%. 

Curves  should  be  marked  with  the  names  appearing  at 
the  heads  of  the  columns  from  which  their  ordinates  were 
taken.  The  curves  from  columns  2  and  5  should  be  to 
the  same  scale  of  ordinates.  Those  from  columns  6,  7,  8, 
and  9  will  all  have  the  same  scale  of  ordinates,  which  will 
be  per  cents,  and  should  run  from  o  to  100%. 

There  will  thus  be  four  scales  of  ordinates,  and  they 
should  be  marked  respectively,  "Watts  or  Volt  Amperes," 
"Volts,"  "Amperes,"  and  "Per  Cent."  On  the  margin 
state  the  name  and  size  of  the  machine,  and  the  date  of 
test. 

123.  Exp.  28.  Complete  Test  of  a  1  H.P.,  Three-Phase 
Induction  Motor,  run  on  a  Single-phase  Circuit  Through  a 
Condenser-Compensator.  —  The  function  of  the  condenser- 
compensator  was  discussed  in  §  67.  The  arrangement  of 
apparatus  is  shown  in  Fig.  182.  Another  wattmeter  and 


Motor 


Fig.  182. 


another  ammeter  may  be  used  to  alternate  with  those 
shown  to  secure  greater  accuracy  in  the  lower  ranges  if 
considered  advisable. 

The  same  absorption  dynamometer  is  used  as  in  Exp. 
27  ;  and  the  directions  there  given  for  taking  observations, 
and  for  calculating  and  plotting  results,  should  be  followed 
with  the  following  exceptions  :  take  readings  at  2-lb.  steps, 


244  ALTERNATING-CURRENT    MACHINES. 

since  the  motor  is  half  the  size  of  the  other ;  since  this  is 
single-phase,  the  wattmeter  readings  go  direct  in  column 
2,  and  the  products  of  the  volts  by  the  amperes  go  direct 
in  column  5. 

It  may  be  found  that  the  capacity  of  the  condenser- 
compensator  has  been  so  proportioned  that  in  plotting 
the  results,  curves  2  and  5,  and  also  7  and  8,  will  be  nearly 
coincident,  and  that  curve  6  is  practically  a  straight  hori- 
zontal line. 

124.  Exp.  29.  Methods  of  Synchronizing.  —  Synchro- 
nous motors  and  also  converters  must  be  synchronized 
before  being  connected  to  the  mains  from  which  they 
receive  their  power.  There  are  a  number  of  ways  of 
doing  this,  of  which  the  best  depends  upon  attendant  cir- 
cumstances, (a)  The  motor  and  generator  may  be  elec- 
trically connected  while  at  rest,  and  the  latter  started  up 
slowly,  the  motor  —  not  loaded  —  then  starting  up  and 
running  synchronously,  (b)  The  field  circuit  of  the  motor 
may  be  left  open,  and  the  armature  started  up  —  without 
load — as  an  induction  motor  until  near  synchronism,  and 
the  field  switch  then  closed.  In  large  machines  this 
endangers  the  insulation  of  the  field  coils,  (c)  The  arma- 
ture may  be  brought  to  speed  mechanically,  either  by  a 
small  direct  connected  induction  motor  or  by  a  belt  from 
some  moving  pulley,  (d]  In  converters  the  machine  can 
be  started  and  brought  to  speed  from  the  direct-current 
end  like  a  direct-current  motor,  if  there  be  direct  cur- 
rent available.  This  requires  a  starting-box  and  a  field 
rheostat. 

The  two  convenient  methods  for  synchronizing  the 
I  K.  w.  three-phase  converter  are  (b}  and  (d) ;  the  former 


TESTS. 


245 


will  be  practiced  in  Exp.  30,  the  latter  is  the  subject  for 
the  present  experiment. 

Arrange  the  apparatus  as  shown  in  Fig.  183.  At 
starting,  the  field  coils  of  the  converter  must  be  excited 
from  the  source  of  direct  current,  but  when  running  as 
a  converter  the  machine  must  be  excited  from  its  own 
brushes.  This  >  necessitates  the  two  switches,  a  and  b. 
These  switches  must  not  be  both  open  at  once,  at  least 
while  the  machine  is  running  from  the  direct-current  end ; 
and  if  they  are  not  rightly  connected  the  direct-current 


Main  Switch 


3  Phase 

67.6  V. 
40  oj 


16  c.  p. 

r\ 

'r\ 

Synchronizing 
Lamps 


J 


Fig.  183. 

source  will  be  short-circuited  when  they  are  both  closed  at 
once.  It  is  best,  after  the  set-up  is  made,  to  test  across 
the  switches  with  a  voltmeter.  The  switch  must  not  be 
closed  if  any  pressure  shows  across  the  gap  it  is  intended 
closing. 

When  the  connections  have  been  properly  made,  open 
the  main  switch  and  switch  -a,  close  switch  b  and  the 
switch  from  the  direct-current  source  of  supply,  and  start 
the  machine  up  as- a  direct-current  motor.  When  the 
starting-box  is  completely  on,  first  close  a,  then  open  b. 


246  ALTERNATING-CURRENT   MACHINES. 

Then  manipulate  the  field  rheostat  until  the  machine 
reaches  synchronism.  The  synchronizing  lamps  will  all 
be  dark  at  once  when  the  machine  is  in  step.  When 
the  periods  of  darkness  become  quite  long,  say  several 
seconds,  the  main  switch  can  be  closed,  the  switch  from 
the  direct-current  source  be  opened,  and  the  machine  will 
be  running  as  a  self-excited  converter. 

If  all  the  lamps  do  not  get  dark  at  once,  but  two  stay 
lighted  while  the  other  is  dark,  the  generator  currents  are 
in  such  directions  as  to  tend  to  reverse  the  direction  of 
rotation  of  the  converter  armature.  Two  of  the  leads  in 
the  alternating-current  circuit  should  then  be  transposed. 
It  might  here  be  noticed  that  if  an  inverted  converter  be 
used  as  a  source  of  alternating  currents,  and  it  be  desired 
to  synchronize  another  converter  with  it  as  described 
above,  and  if  an  Edison  three-wire  system  be  the  common 
source  of  direct-current  supply  to  these  machines,  then 
care  must  be  taken  that  both  converters  are  connected  to 
the  same  side  of  the  system.  If  they  be  connected 
to  opposite  sides,  then  when  the  machines  are  in  step 
there  is  a  pressure  of  117  volts  across  the  main  switch, 
and  closing  the  latter  would  naturally  cause  the  blowing 
of  some  fuse. 

125.  Exp.  30.  Variation  of  Lag  or  Lead  of  Current 
in  a  Three-Phase  i  K.W.  Synchronous  Motor.  —  Arrange 
the  apparatus  as  in  Fig.  184.  Either  the  3  or  the  15 
ampere  ammeter  can  be  put  in  circuit,  according  to  the  load. 
Remember  that  in  starting  up  there  will  be  an  excessive 
flow  of  current. 

The  direct-current  brushes  on  the  machine  may  be 
removed  for  this  experiment,  Synchronize  the  motor  by 


TESTS. 


letting  it  run  as  an  induction  motor  till  near  synchronism. 
Then  close  the  field  switch,  having  first  adjusted  the 
rheostat,  so  that  about  normal  field  current  will  flow.  The 
machine  will  fail  to  go  into  step  if  this  adjustment  is  not 
made.  Premature  closing  of  the  field  switch  is  also  a 
cause  of  failure  to  synchronize.  It  is  easy  to  tell  if  the 
machine  goes  into  step  or  not.  If  it  does,  the  current  in 
the  armature  circuit  goes  down ;  if  it  does  not,  the  machine 
slows  down  and  even  stops,  and  the  current  goes  up. 

(a)   When  the  motor  is  synchronized,   reduce  the  field 
current  as  far  as  possible,  without  losing  step.     In  some 


3  Phase 


Pig.  184. 


machines  it  may  be  reduced  to  zero,  the  residual  magnet- 
ism affording  enough  field  to  keep  the  armature  syn- 
chronized. From  this  point  increase  the  field  current  by 
suitable  steps  to  the  maximum  allowable  current,  or  until 
the  machine  loses  synchronism.  At  each  step  note  the 
field  current  and  the  armature  current.  Then  from  the 
maximum  field  current,  decrease  to  the  minimum  by 
the  same  steps,  taking  readings  again  of  the  two  ammeters. 
Due  to  magnetic  retentivity  the  two  curves  will  not 
coincide. 

Plot  the  two  curves  on  one  paper,  using  field  currents 
as  abscissae,  and  armature  currents  as  ordinates. 


248  ALTERNATING-CURRENT   MACHINES. 

That  excitation  at  which  the  no-load  armature  current  is 
a  minimum,  is  called  the  normal  excitation  of  a  syn- 
chronous machine. 

(b)  Repeat  the  foregoing,  save  that  a  strap-brake  load  is 
applied  to  the  motor.  This  load  should  be  adjusted  to 
about  75%  of  full  load,  and  left  constant.  It  will  be 
found  that  the  motor  will  not  submit  to  so  wide  a 
range  of  field  currents  when  loaded  as  when  running 
light. 

Plot  these  curves  on  the  same  sheet. 

126.  Exp.  31.  Commercial  Efficiency  of  a  Synchron- 
ous Motor.  — The  same  arrangement  of  the  same  appa- 
ratus is  here  used  as  in  Exp.  27,  save  that  it  is  applied  to 
a  three-phase  synchronous  motor,  whose  field  coils  must 
be  separately  excited  from  a  direct-current  source,  and 
with  a  suitable  rheostat  in  series. 

Synchronize  the  motor  by  the  method  of  Exp.  30,  being 
careful  that  the  excessive  starting  current  does  not  pass 
through  the  coil  of  the  low-reading  wattmeter.  When  the 
armature  is  in  step,  adjust  the  field  rheostat  to  give  normal 
excitation  ;  i.e.,  so  that  the  armature  current  is  a  minimum. 
This  adjustment  must  not  be  changed  during  the  experi- 
ment. The  frequency  should  be  kept  constant,  and  the 
voltage  also  if  possible  ;  if  not,  account  should  be  taken  of 
its  fall,  and  a  voltage  curve  drawn  on  the  same  sheet  as  the 
efficiency  curve. 

Load  the  motor  by  the  strap-brake  shown  in  Exp.  27, 
increasing  by  suitable  steps  from  zero  till  the  motor  falls 
out  of  step.  At  each  step  read  the  wattmeter  and  the 
two  spring  balances.  Repeat  the  experiment  three  times, 
each  time  stopping  at  the  same  points  on  the  larger  spring 


TESTS.  249 

balance,  so  that  the  other  values  can  be  averaged,  reducing 
errors  of  observation. 

Plot  a  curve  with  watts  output  and  per  cent  of  load  as 
abscissae,  and  per  cent  efficiency  as  ordinates. 

127.  Exp.  32.     Curves   of    Current   and    Power-Factor 
of   a  Synchronous  Motor,   with  (a)  Super-Excitation,  and 
(b)  Sub-Excitation.  --  The  arrangement  of  apparatus  is  that 
of  Exp.  27,  applied  to  the  synchronous  motor. 

For  the  first  part  of  the  experiment,  the  fields  should  be 
excited  by  a  current  about  50%  greater  than  the  normal 
field  current,  and  for  the  second  part  by  a  current  about 
50%  less.  The  frequency  and  the  voltage  should  be  kept 
constant. 

For  each  part  load  the  motor  with  the  strap-brake,  in- 
creasing from  zero  by  suitable  steps  till  the  armature  falls 
out  of  synchronism.  At  each  step  take  readings  of  the 
ammeter,  voltmeter,  and  wattmeter,  as  well  as  of  the  two 
spring  balances.  Repeat  each  part  three  times,  averaging 
the  results. 

Tabulate  the  results  under  columns  headed  "  Watts  Out- 
put," "Watts  Input,"  "Volt-amperes  Input,"  and  "Power- 
factor." 

Plot  on  one  paper  the  curves  for  the  super-excited 
condition,  making  watts  output  the  common  abscissae,  and 
armature  currents  and  power-factors  respectively  the 
ordinates. 

Plot  on  another  sheet  similar  curves  for  the  sub-excited 
condition. 

128.  Exp.   33.     External  Characteristic  of  a  Converter, 
A.C.   to  D.C.   with  Self -Excitation.  —  Arrange   apparatus 
as  in  Fig.  183,  Exp.  29.     When  the  converter  is  running 


2i>0          ALTERNATING-CURRENT   MACHINES. 

from  the  alternating  end,  and  free  from  the  source  of 
direct-current  supply,  adjust  the  field  rheostat  to  that 
point  that  gives  a  minimum  armature  current  at  no  load. 
If  this  point  is  not  already  known,  it  will  be  necessary 
to  put  the  i5-ampere  ammeter  in  one  of  the  alternating- 
current  mains  to  determine  it. 

When  the  above  conditions  are  fulfilled,  load  the  direct- 
current  end  of  the  converter  with  lamps,  from  o  up  to  50% 
overload  (say  15  amperes),  by  steps  of  about  one  ampere 
each.  At  each  step  take  readings  of  the  armature  current 
and  the  terminal  pressure,  using  the  standard  direct-current 
instruments  for  the  purpose. 

Plot  a  curve  with  armature  currents  as  abscissae,  and 
terminal  pressure  as  ordinates. 

129.  Exp.  34.  Efficiency  of  a  Converter  from  A.C.  to 
D.C. — The  arrangement  of  apparatus  is  that  of  the  last 
experiment  with  the  addition  of  a  wattmeter  suitably  con- 
nected in  the  alternating-current  mains  to  measure  the 
power  input.  In  fact,  all  the  necessary  data  for  Exp.  33 
are  incidentally  secured  in  the  course  of  this  experiment. 

Run  and  excite  converter  as  in  the  last  experiment. 
The  frequency  and  the  voltage  should  be  kept  constant. 
The  direct-current  end  is  to  be  loaded  with  lamps  by  suit- 
able steps  from  o  to  50%  overload.  The  watts  input  can 
be  determined  from  the  wattmeter  reading,  the  watts  out- 
put from  the  product  of  the  voltmeter  and  the  ammeter 
reading. 

Plot  an  efficiency  curve. 

NOTE.  —  The  brush  friction  of  the  direct-current  brushes  may  be  so 
great  that  the  converter  cannot  be  synchronized  by  the  method  of  starting 
as  an  induction  motor,  the  slip  being  so  great  as  to  prevent  its  picking  up 


TESTS.  251 

when  the  field  circuit  is  closed.  In  such  case  either  the  direct-current 
brushes  must  be  temporarily  removed,  or  the  machine  must  be  synchronized 
by  some  of  the  other  methods  given  in  Exp.  29.  If  the  converter  hunts  so 
badly  as  to  interfere  with  the  instrument  readings,  it  may  be  because  the 
direct-current  brushes  are  not  in  proper  adjustment,  and  the  selection  of  a 
better  commutating  plane  will  remedy  the  trouble. 

130.   Exp.  35.   Efficiency  of  an  Inverted  Converter.  — 

Since  it  is  inconvenient  to  put  a  variable,  non-inductive, 
balanced  load  on  a  three-phase  circuit,  the  single-phase 
rings  will  be  used  in  this  experiment. 

The  arrangement  of  apparatus  requires  a  direct-current 
ammeter  and  voltmeter  on  the  D.C.  end,  and  a  wattmeter 
on  the  A.C.  end.  Or  two  wattmeters  can  be  used  if  avail- 
able. The  converter  is  started  from  the  D.C.  end  by 
means  of  a  starting-box,  the  field  rheostat  is  adjusted  until 
the  armature  is  running  at  its  rated  speed.  This  speed 
must  be  kept  constant  during  the  test  by  manipulating  the 
rheostat.  A  non-inductive  load  is  applied  to  the  A.C.  end, 
increasing  by  suitable  steps  from  o  to  50%  overload. 

Plot  an  efficiency  curve  and  an  external  characteristic 
curve.  The  latter  will  approximate  a  straight  horizontal 
line,  being  much  better  than  that  secured  in  Exp.  33,  be- 
cause, in  this  case,  the  field  current  is  unaffected  by  any 
drop  of  voltage  in  the  armature. 


NDEX. 


[The  figures  refer  to  page  numbers.] 


Admittance  of  circuit,  45. 
Ageing  of  iron,  123. 
Air-blast  transformers,  128. 
All-day  efficiency,  139. 
Alternating  current,  definition  of,  I. 
Alternations,  definition  of,  i. 
Alternators,  Bullock,  89. 

core  loss  of,  238. 

efficiency  of,  72. 

external  characteristic  of,  235. 

field  compounding  of,  235. 

General  Electric  Co.'s,  76, 78, 87. 

inductor,  80. 

load  losses  of,  73. 

parallel  running  of,  168. 

revolving  field,  85. 

saturation    curve    of,  full-load, 

236. 
no-load,  235. 

single-phase,  57. 

Stanley,  81. 

Warren,  85. 

Westinghouse,  76,  91. 
Aluminum  line  wire,  189. 
Angle  of  lag  or  lead,  10. 
Average  values  of  pressure  and  cur- 
rent, 8. 

Apparent  resistance,  25,  45. 
Armature,  E.M.F.  generated  in,  62. 

inductance,  68. 

reaction,  67. 


Armature : 

windings,  65. 
Auto-transfonner,  94. 

connections  of,  121. 

Brake,  Prony,  241. 
Bullock  alternators,  89. 

Calculation  of  leakage  flux,  108. 

resulting  impedance,  50,  224. 
Capacity,  measurement  of,  213. 

of  circuit,  45. 

of  condenser,  29. 
formula  for,  31. 

of  transmission  line,  193. 

reactance,  40,  45. 

unit  of,  30. 

Centrifugal  clutch  pulley,  152. 
Characteristic  of  alternator,  235. 

of  converter,  249. 

of  inverted  converter,  251. 
Chemical  solution  to  detect  current. 

204. 

Choke  coils,  28. 
Circuits,  time  constant  of,  21. 

with  ^,  Z,  and  C,  40. 
Coefficient  of  leakage,  71. 

of  saturation,  70. 

of  self-induction,  16. 
Combined  method,  measuring  power, 
218. 


254 


INDEX. 


Commercial   efficiency    of    synchro- 
nous motor,  248. 
Compensated  winding,  78. 
Compensators,  94. 

connections  of,  121. 
Composite  winding,  74,  76. 
Compounding   curve    of   alternator, 

235- 

Condenser,  hydraulic  analogy,  37. 
compensator,  1 54. 
capacity  of,  29. 

formula  for,  31.      - 
electrolytic,  31. 
Condensers,  29. 
in  parallel,  32. 
in  series,  33. 

Conductance  of  circuit,  46. 
Connections  of  transformers,  115. 
Constant  current  transformers,   130, 

potential,  regulation  for,  57. 
Converter,  169. 

armature  heating,  175. 

reaction,  177. 
current  relations  in,  -173. 
efficiency  of,  250. 
E.M.F.  relations  in,  171. 
external  characteristic  of,  249. 
inverted,  170. 

efficiency  of,  251. 
external  characteristic  of ,  2  5 1 . 
regulation  of,  179. 
•  starting  of,  177. 
Cooling  of  transformers,  128. 
Copper  loss  in  transformers,  104. 

weight  of,  for  lines,  197. 
Core  flux  in  transformer,  92,  125. 
loss  of  alternator,  74. 

measurement  of,  238. 
of  transformer,  99. 

measurement  of,  231. 
type  transformer,  92,  125. 


Counter  E.M.F.  of  self-induction,  26. 
Current  and  pressure  relations  : 
in  a  condenser,  39. 
transformer,  determination  of, 

231. 

average  value  of,  8. 
effective  value  of,  7. 
flow,  formula  for,  42. 
instantaneous  value  of,  4. 
lag  or  lead  of,  10. 
magnetic  energy  of,  23. 
produced  by  harmonic  E.M.F., 

24. 
Curve,  saturation,  full  load,  236. 

no  load,  235. 
sine,  4. 

form-factor  of,  9. 
Curves  from  transformer,  231. 

of  current  and  power-factor  in 

synchronous  motor,  249. 
E.M.F.,  actual,  6. 
distortion  of,  9. 
determination  of,  205,  208. 
field  compounding  of  alterna- 
tor, 235. 
Cycle,  definition  of,  i. 

Decay  of  current  in  circuit,  21. 
Definitions  of  terms,  44. 
Delta  or  mesh  connection,  61. 

of  transformers,  119. 
Design  of  transformer,  133. 
Dielectric  hysteresis,  31. 
Dielectrics  for  condensers,  30. 
Distribution  constant,  64. 

Eddy  current  loss,  99. 
Effective  values  of  current  and  pres- 
sure, 7, 

Efficiency,  all  day,  105. 
of  alternator,  72. 


INDEX. 


255 


Efficiency : 

of  converter,  250. 
of  inverted  converter,  251. 
of  synchronous  motor,  248. 
of  transformer,  104,  05,  139. 

measurement  of,  228. 
E.M.F.,  counter,    of   self-induction, 

26. 

generated  in  armature,  62. 
E.M.F.,  wave,  shape  of,  3,  6. 
determination  of,  205. 
E.M.F?§  in  series,  46. 
Energy  of  a  started  current,  23. 
Equivalent  A',  X,  and  Z  of  trans- 
former, 97. 

leakage  inductance,  108. 
Exact  solution  of  transformer,  in. 
Exciting  current  of  transformer,  94, 

103. 
External  characteristic  of  alternator, 

235. 

converter,  249. 
inverted  converter,  251. 

Farad,  definition  of,  30. 
Field  compounding  curve  of  alter- 
nator, 235. 

rotating,  141. 

Flux  density  in  transformers,  95. 
Form-factor,  9. 

determination  of,  206. 
Formula  for  current  in  any  circuit, 

42. 
Four-phase  currents,  n. 

systems,  59. 
Frequency  changers,  157. 

definition  of,  I. 

determination  of,  2. 
Frequencies,    for    power    transmis- 
sion, 183. 

standard,  2. 


Full-load  saturation  curve,  70. 
determination  of,  236. 

General   Electric    Co.'s    alternator, 

76,  78,  87. 
regulator,  181. 
induction  motor,  146. 
transformer,  125,  132. 
Growth   of   current    in    condensive 

circuit,  34. 
in  inductive  circuit,  20. 

Harmonic  E.M.F.,  current  produced 
by,  24. 

shadowgraph,  3. 
Henry,  definition  of,  16. 
Hydraulic  analogy  of  condenser,  37. 
Hysteresis,  dielectric,  31. 

loss  in  alternators,  73. 

transformers,  100. 

Impedance,  definition  of,  25. 

of  circuit,  45,  224—228. 

synchronous,  69. 

measurement  of,  237. 
Impedances  in  parallel,  226. 

in  series,  224. 
Inductance,  measurement  of,  211. 

mutual,  measurement  of,  216. 

of  circuit,  45. 

of  transmission  lines,  191. 

self,  described,  15. 

unit  of  self,  16. 

variation  with  load  of,  214. 
Inductances,  practical  values  of,  18, 
Induction  motors,  142.. 

behavior  of,  149. 

General  Electric  Co.'s,  146. 

single  phase,  154. 

slip  of,  144. 

speed  regulation,  157. 


256 


INDEX. 


Induction  motors  : 

starting  of,  150. 

tests  of,  239,  243. 

Wagner,  155. 

Westinghouse,   143. 
Inductive  reactance,  25,  45. 
Inductor  alternators,  80. 
Instantaneous  values  of  current  and 

pressure,  4. 
Inverted  converter,  170. 

Lag  of  current,  i  o. 
Lead  of  current,  10. 
Leakage  coefficient,  71. 

flux,  1 06. 

inductance,  108. 
Lighting  transformers,  122. 
Line  capacity,  193. 

constants  (Table),  194. 

inductance,  191. 

loss,  curves  of,  196. 

resistance,  190. 

wire,  aluminum,  189. 
Linkages  defined,  16. 
Load  losses  in  alternator,  73. 
in  transformer,  99. 

measurement  of,  229. 
Logarithmic  change  of  current,  20. 
Losses  in  synchronous  machines,  73. 

in  transformers,  99. 

in  transmission  lines,  186. 

Magnetic  energy  of  current,  23. 
Magnetizing  current  of  transformer, 

102. 

Magnitude  of  self-inductance,  19. 
Measurement  of  capacity,  213. 

of  core  loss  in  transformer,  231. 
of  core  loss  in  alternator,  238. 
of     efficiency    of    transformer, 
228. 


Measurement : 

of  load  losses  in   transformer, 

229. 

of   mutual   induction,  216. 
of  transformer  coils,  233. 
of  power,  polyphase,  219. 
of  power,  single  phase,  217. 
of   regulation    of    transformer, 

228. 
of    resulting     impedance,    224, 

226,  228. 

of  self-inductance,  211. 
Mershon  balance,  206. 
Mesh  or  delta  connection,  61. 

of  transformers,  119. 
Methods   of  connecting   transform- 
ers, 115. 

of  synchronizing,  244. 
Microfarad,  definition  of,  30. 
Monocyclic  system,  156. 
Motor,  induction,  142. 
behavior  of,  149. 
General  Electric  Co.'s,  146. 
measurement     of    efficiency, 

239»  243- 
single-phase,  154. 
speed  regulation,  157. 
starting  of,  150. 

induction,    treatment  by  trans- 
former method,  147. 
Wagner,  155. 
Westinghouse,  143. 
starters,  General  Electric  Co.'s, 

I5i- 

Westinghouse,  151. 
synchronous,  158. 

measurement     of     efficiency, 

248. 
Mutual  induction,  measurement  of, 

216. 
in  transformer  coils,  233. 


INDEX. 


257 


Natural  draft  transformers,  128. 
No-load  saturation  curve,  70. 

determination  of,  235. 
Number  of  phases  for  transmission, 

.      189- 

Oil-cooled  transformers,  129. 
Operation  of  induction  motors,  143. 
Operative    range     of  '  synchronous 
motors,  161. 

Parallel  circuits,  226. 
Parallel  running  of  alternators,  168. 
Parallel-series  circuits,  227. 
Parallelogram  of  E.M.F.'1^  27,  48. 
Peculiarities  of  A.C.  circuits,  203. 
Phase,  10. 

relations  in  condensive  circuits, 

37- 

splitters,  153. 
Phases,  number  of,  for  transmission, 

189. 

Polygon  of  admittances,  53. 
of  E.M.F's,  49. 
of  impedances,  50. 
Polyphase  alternators,  58. 
currents,  12. 

power,  measurement  of,  219. 
Power     curves,     determination     of, 

209. 

factor,  definition  of,  14. 
in  A.C.  circuits,  12. 

measurement  of,  217-224. 
transmission,  182. 
Power  transmission,  frequency  for, 

183. 

voltage  for,  185. 
phases  for,  189. 

Practical  values  of  inductances,  18. 
Pressure   and   current    relations    in 
condenser,  39. 


Pressure   and   current    relations  in 
transformer,  determination  of, 

231. 

average  value  of,  8. 
curves,  actual,  6. 

causes  of  distortion,  5. 
determination  of,  205,  208. 
effective  value  of,  7. 
for  transmission,  185. 
instantaneous  value  of,  4. 
Primary  of  transformer,  92. 
Prony  brake,  241. 

Quarter-phase  currents,  4. 
systems,  59. 

Ratio  of  transformation,  92. 

in  induction  motor,  148. 
Reactance  of  any  circuit,  45. 
of  condensive  circuit,  40. 
of  inductive  circuit,  25. 
synchronous,  69. 

measurement  of,  237. 
Regulation  for  constant    potential, 

74- 

of  converters,  179. 
of  transformers,  106,  139. 

measurement  of,  228. 
Regulator,  General     Electric   Co.'s, 

181. 

Stillwell,  1 80. 
Relations  of  current  and  pressure  in 

condenser,  39. 
in  transformer,  determination 

of,  231. 

of  E.M.F.'s,  in  converters,  173^ 
Resistance,  apparent,  25,  45. 

of  inductive  circuits,  25,  44. 
of  line  wire,  190. 
Resonance,  42. 
Revolving  field  alternators,  85. 


258 


INDEX. 


Rotary  converter,  see  Converter. 
Rotating  magnetic  field,  141. 
Rotor,  definition  of,  142. 

Saturation,  coefficient,  70. 
curves,  70. 
of  alternator,  full-load,  236. 

no-load,  235. 
Scott  transformer,  118. 
Secondary  of  transformer,  92. 
Self-inductance,  counter  E.M.F.  of, 
26. 

described,  15. 
measurement  of,  211. 
unit  of,  1 6. 
Series  circuits,  224. 
Series-parallel  circuits,  227. 
Shadowgraph,  harmonic,  3. 
Shape  of   current  wave,  determina- 
tion of,  208. 
of  E.M.F.  wave,  3. 

determination  of,  205,  209. 
Shell  type  transformer,  92,  123. 
Simultaneous     curves    from    trans- 
former, 231. 
Sine  curve,  4. 

form  factor  of,  9. 
Single-phase  alternators,  57. 
current,  n. 
induction  motor,  154. 

test,  243. 

power,  measurement  of,  217. 
Sinusoid,  4. 

Slip  of  induction  motors,  144. 
Skin  effect,  191. 

Solution  for  detecting  currents,  204. 
Squirrel -cage     motors,    starting    of, 

150. 

rotor  defined,  143. 
Standard  frequencies,  2. 
Stanley  alternator,  81. 


Stanley  transformer,  127. 
Star  or  Y  connection,  60. 

of  transformers,  119. 
Started  current,  magnetic  energy  of, 

23- 

Starting  of  induction  motors,  150. 
of  synchronous  motors,  155. 
Stator  defined,  142. 
Step-up    and   step-down    transform- 
ers, 93. 

Stillwell  regulator,  180 
Strap  brake,  241. 
Susceptance  of  circuit,  46. 
Synchronizer,  167. 
Synchronizing,  methods  of,  244. 
Synchronous  motors,  158. 
hunting  of,  1 63. 
measurement     of     efficiency, 

248. 

operative  range,  161. 
starting  of,  165. 
variation  of  lag  or  lead,  246. 
of  current  and  power-factor, 

249. 
reactance,  69. 

measurement  of,  237. 

Table  of  line  constants,  194. 
Temperature.effect  on  core  loss,  103. 
Test  of  induction  motor,  239. 

with  condenser  compensator, 

243- 

Three-ammeter  method  for  measur- 
ing power,  218. 
voltmeter  method,  217. 
Three-phase  induction   motor  test, 

239- 

power  measurements,  221. 
systems,  60. 

transformations,  119,  234. 
Time  constant  of  circuit,  2j, 


INDEX. 


259 


Transformation,  ratio  of,  92. 
Transformer,  air  blast,  129. 
connections  of,  115,  234. 
constant  current,  130. 
cooling  of,  128. 
definition,  92. 
design  of,  133. 
efficiency,  104,  115,  139, 

measurement  of,  228. 
exact  solution  of,  in. 
flux  in,  94. 
for  lighting,  F22. 
General  Electric  Co.'s,  125,  132. 
losses,  99. 
measurement  of  core  losses,  231. 

load  losses,  229. 

mutual  induction  in,  234. 
method  of  treatment  for  induc- 
tion motors,  147. 
natural  draft,  128. 
oil  cooled,  129. 
regulation  of,  106,  139. 

measurement  of,  228. 
Scott,  ii  8. 

simultaneous  curves  from,  311. 
Stanley,  127. 
Wagner,  123. 
water-cooled,  129. 
Westinghouse,  126. 
Transmission  of  power,  182. 
lines,  capacity  of,  1 93. 

inductance  of,  191. 

resistance  of,  190. 

table  of  constants  for,  194. 
losses  in,  186. 

curves  of,  196. 
Triangle  of  E.M.F?s>  25,  40. 


Two-phase  currents,  n. 

power  measurements,  219. 

systems,  59. 
Type  A.  O.  transformers,  127. 

H  transformers,  i  25. 

M  transformers,  123. 

O.D.  transformers,  126. 

Variation  in  synchronous  motor  of 
lag  or  lead,  246. 

in    current    and   power   factor, 
249. 

of  inductance  with  load,  214. 
Vibrating  filament,  205. 
Voltage,  average  value,  8. 

curves  of  actual,  6. 

effective  value  of,  7. 

generated  in  armature,  62. 

Wagner  induction  motor,  155. 

transformer,  123. 
Warren  alternator,  85. 
Water-cooled  transformers,  129. 
Wattmeters   in   polyphase    circuits, 

219. 
Wave-shape,  3. 

causes  of  distortion  of,  5. 

determination  of,  205,  208. 

form  factor  of,  9. 
Weight  of  copper  for  lines,  197. 
Westinghouse  alternator,  76,  91. 

induction  motor,  143. 

transformer,  126. 

Y-connection,  60. 

of  compensators,  121. 
of  transformers,  119. 


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